data_source stringclasses 1 value | prompt listlengths 1 1 | ability stringclasses 1 value | reward_model dict | extra_info dict | level stringclasses 1 value | problem_id stringclasses 1 value |
|---|---|---|---|---|---|---|
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:1 \\qquad \\textbf{(... | math | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance | easy-99-without-guidance |
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