Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
40,300 | If
\[\sin x + \cos x + \tan x + \cot x + \sec x + \csc x = 7,\]then find $\sin 2x.$ | 22 - 8 \sqrt{7} | 27.34375 |
40,301 | Find the phase shift of the graph of $y = \sin (3x - \pi).$ | -\frac{\pi}{3} | 0.78125 |
40,302 | Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$. | 628 | 92.96875 |
40,303 | Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\] | 63 | 94.53125 |
40,304 | Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\] | 0 | 71.09375 |
40,305 | Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point. Then there exists a constant $k$ so that
\[PA^2 + PB^2 + PC^2 = k \cdot PG^2 + GA^2 + GB^2 + GC^2.\]Find $k.$ | 3 | 96.09375 |
40,306 | If angle $A$ lies in the second quadrant and $\sin A = \frac{3}{4},$ find $\cos A.$ | -\frac{\sqrt{7}}{4} | 86.71875 |
40,307 | The real numbers $a$ and $b$ satisfy
\[\begin{pmatrix} 2 \\ a \\ -7 \end{pmatrix} \times \begin{pmatrix} 5 \\ 4 \\ b \end{pmatrix} = \mathbf{0}.\]Enter the ordered pair $(a,b).$ | \left( \frac{8}{5}, -\frac{35}{2} \right) | 96.875 |
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