problem stringlengths 10 5.15k | answer dict |
|---|---|
A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$. | {
"answer": "797",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the measures of $\angle ABC, \angle BCA,$ and $\angle XOY$ are in the ratio $13 : 2 : 17,$ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Diagram
[asy] /* Made by MRENTHUSIASM */ size(375); pair A, B, C, O, G, X, Y; A = origin; B = (1,0); C = extension(A,A+10*dir(585/7),B,B+10*dir(180-585/7)); O = circumcenter(A,B,C); G = centroid(A,B,C); Y = intersectionpoint(G--G+(100,0),B--C); X = intersectionpoint(G--G-(100,0),A--scale(100)*rotate(90)*dir(O-A)); markscalefactor=3/160; draw(rightanglemark(O,G,X),red); dot("$A$",A,1.5*dir(180+585/7),linewidth(4)); dot("$B$",B,1.5*dir(-585/7),linewidth(4)); dot("$C$",C,1.5N,linewidth(4)); dot("$O$",O,1.5N,linewidth(4)); dot("$G$",G,1.5S,linewidth(4)); dot("$Y$",Y,1.5E,linewidth(4)); dot("$X$",X,1.5W,linewidth(4)); draw(A--B--C--cycle^^X--O--Y--cycle^^A--X^^O--G^^circumcircle(A,B,C)); [/asy] ~MRENTHUSIASM | {
"answer": "592",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(n)$ and $g(n)$ be functions satisfying \[f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}\]and \[g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}\]for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$. | {
"answer": "258",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD=BC=333$. The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$, and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$. Find $PQ$.
Diagram
[asy] /* Made by MRENTHUSIASM */ size(300); pair A, B, C, D, A1, B1, C1, D1, P, Q; A = (-250,6*sqrt(731)); B = (250,6*sqrt(731)); C = (325,-6*sqrt(731)); D = (-325,-6*sqrt(731)); A1 = bisectorpoint(B,A,D); B1 = bisectorpoint(A,B,C); C1 = bisectorpoint(B,C,D); D1 = bisectorpoint(A,D,C); P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); draw(anglemark(P,A,B,1000),red); draw(anglemark(D,A,P,1000),red); draw(anglemark(A,B,Q,1000),red); draw(anglemark(Q,B,C,1000),red); draw(anglemark(P,D,A,1000),red); draw(anglemark(C,D,P,1000),red); draw(anglemark(Q,C,D,1000),red); draw(anglemark(B,C,Q,1000),red); add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$D$",D,1.5*dir(D),linewidth(4)); dot("$P$",P,1.5*NE,linewidth(4)); dot("$Q$",Q,1.5*NW,linewidth(4)); draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); label("$500$",midpoint(A--B),1.25N); label("$650$",midpoint(C--D),1.25S); label("$333$",midpoint(A--D),1.25W); label("$333$",midpoint(B--C),1.25E); [/asy] ~MRENTHUSIASM ~ihatemath123 | {
"answer": "242",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $w = \dfrac{\sqrt{3} + i}{2}$ and $z = \dfrac{-1 + i\sqrt{3}}{2},$ where $i = \sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \cdot w^r = z^s.$ | {
"answer": "834",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ordered pairs of integers $(a, b)$ such that the sequence\[3, 4, 5, a, b, 30, 40, 50\]is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. | {
"answer": "228",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,b,c,d,e,f,g,h,i$ be distinct integers from $1$ to $9.$ The minimum possible positive value of \[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\] can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | {
"answer": "289",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega.$ Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A,$ $\omega_B,$ and $\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\sqrt{a} - \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C, W, WA, WB, WC, X, Y, Z; A = 18*dir(90); B = 18*dir(210); C = 18*dir(330); W = (0,0); WA = 6*dir(270); WB = 6*dir(30); WC = 6*dir(150); X = (sqrt(117)-3)*dir(270); Y = (sqrt(117)-3)*dir(30); Z = (sqrt(117)-3)*dir(150); filldraw(X--Y--Z--cycle,green,dashed); draw(Circle(WA,12)^^Circle(WB,12)^^Circle(WC,12),blue); draw(Circle(W,18)^^A--B--C--cycle); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$\omega$",W,1.5*dir(270),linewidth(4)); dot("$\omega_A$",WA,1.5*dir(-WA),linewidth(4)); dot("$\omega_B$",WB,1.5*dir(-WB),linewidth(4)); dot("$\omega_C$",WC,1.5*dir(-WC),linewidth(4)); [/asy] ~MRENTHUSIASM ~ihatemath123 | {
"answer": "378",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ellina has twelve blocks, two each of red ($\textbf{R}$), blue ($\textbf{B}$), yellow ($\textbf{Y}$), green ($\textbf{G}$), orange ($\textbf{O}$), and purple ($\textbf{P}$). Call an arrangement of blocks $\textit{even}$ if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement \[\textbf{R B B Y G G Y R O P P O}\] is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | {
"answer": "247",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrt n$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
[asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("$A$", A, SW); B=(6,15); label("$B$", B, NW); C=(30,15); label("$C$", C, NE); D=(24,0); label("$D$", D, SE); P=(5.2,2.6); label("$P$", (5.8,2.6), N); Q=(18.3,9.1); label("$Q$", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/asy] | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define \[ S_n = \sum | A \cap B | , \] where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3, \cdots , n \right\}$ with $|A| = |B|$. For example, $S_2 = 4$ because the sum is taken over the pairs of subsets \[ (A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} , \] giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$. Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by 1000. | {
"answer": "245",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a,$ $b,$ $c,$ or $d$ is nonzero. Let $N$ be the number of distinct numerators obtained when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\overline{3636} = \frac{4}{11}$ and $0.\overline{1230} = \frac{410}{3333}.$ Find the remainder when $N$ is divided by $1000.$ | {
"answer": "392",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the $\textit{splitting line}$ of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively, and suppose that the splitting lines of $\triangle ABC$ through $M$ and $N$ intersect at $30^\circ.$ Find the perimeter of $\triangle ABC.$ | {
"answer": "459",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations: \begin{align*} \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\ \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\ \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. \end{align*} Then $\left[ (1-x)(1-y)(1-z) \right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived. | {
"answer": "154",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | {
"answer": "841",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles. | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\ell_A$ and $\ell_B$ be two distinct perpendicular lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than 1 of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8.0 regions when $m=3$ and $n=2$ [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy] | {
"answer": "244",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$.
~ pi_is_3.14 | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a convex quadrilateral with $AB=2$, $AD=7$, and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}$. Find the square of the area of $ABCD$. | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that\[\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.\]Find the least possible value of $a+b.$ | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a polynomial $P(x)$ with integer coefficients such that\[P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\]holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$. | {
"answer": "220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$. | {
"answer": "188",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy] | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | {
"answer": "191",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of all positive integers $m$ such that $\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$ | {
"answer": "012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | {
"answer": "051",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$. | {
"answer": "049",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rhombus $ABCD$ has $\angle BAD < 90^\circ.$ There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to the lines $DA,AB,$ and $BC$ are $9,$ $5,$ and $16,$ respectively. Find the perimeter of $ABCD.$
Diagram
[asy] /* Made by MRENTHUSIASM; inspired by Math Jams. */ size(300); pair A, B, C, D, O, P, R, S, T; A = origin; B = (125/4,0); C = B + 125/4 * dir((3,4)); D = A + 125/4 * dir((3,4)); O = (25,25/2); P = (15,5); R = foot(P,A,D); S = foot(P,A,B); T = foot(P,B,C); markscalefactor=0.15; draw(rightanglemark(P,R,D)^^rightanglemark(P,S,B)^^rightanglemark(P,T,C),red); draw(Circle(O,25/2)^^A--B--C--D--cycle^^B--T); draw(P--R^^P--S^^P--T,red+dashed); dot("$A$",A,1.5*dir(225),linewidth(4.5)); dot("$B$",B,1.5*dir(-45),linewidth(4.5)); dot("$C$",C,1.5*dir(45),linewidth(4.5)); dot("$D$",D,1.5*dir(135),linewidth(4.5)); dot("$P$",P,1.5*dir(60),linewidth(4.5)); dot(R^^S^^T,linewidth(4.5)); dot(O,linewidth(4.5)); label("$9$",midpoint(P--R),dir(A-D),red); label("$5$",midpoint(P--S),dir(180),red); label("$16$",midpoint(P--T),dir(A-D),red); [/asy] ~MRENTHUSIASM | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\{-20,-19,-18,\ldots,18,19,20\},$ such that there is a unique integer $m \not= 2$ with $p(m) = p(2).$ | {
"answer": "738",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.
(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.) | {
"answer": "944",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of subsets of $\{1,2,3,\ldots,10\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\{\mathbf{1},\mathbf{2},5\}$ and $\{1,3,\mathbf{6},\mathbf{7},10\}.$ | {
"answer": "235",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be an equilateral triangle with side length $55.$ Points $D,$ $E,$ and $F$ lie on $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, with $BD = 7,$ $CE=30,$ and $AF=40.$ Point $P$ inside $\triangle ABC$ has the property that \[\angle AEP = \angle BFP = \angle CDP.\] Find $\tan^2(\angle AEP).$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(300); pair A, B, C, D, E, F, P; A = 55*sqrt(3)/3 * dir(90); B = 55*sqrt(3)/3 * dir(210); C = 55*sqrt(3)/3 * dir(330); D = B + 7*dir(0); E = A + 25*dir(C-A); F = A + 40*dir(B-A); P = intersectionpoints(Circle(D,54*sqrt(19)/19),Circle(F,5*sqrt(19)/19))[0]; draw(anglemark(A,E,P,20),red); draw(anglemark(B,F,P,20),red); draw(anglemark(C,D,P,20),red); add(pathticks(anglemark(A,E,P,20), n = 1, r = 0.2, s = 12, red)); add(pathticks(anglemark(B,F,P,20), n = 1, r = 0.2, s = 12, red)); add(pathticks(anglemark(C,D,P,20), n = 1, r = 0.2, s = 12, red)); draw(A--B--C--cycle^^P--E^^P--F^^P--D); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$D$",D,1.5*S,linewidth(4)); dot("$E$",E,1.5*dir(30),linewidth(4)); dot("$F$",F,1.5*dir(150),linewidth(4)); dot("$P$",P,1.5*dir(-30),linewidth(4)); label("$7$",midpoint(B--D),1.5*S,red); label("$30$",midpoint(C--E),1.5*dir(30),red); label("$40$",midpoint(A--F),1.5*dir(150),red); [/asy] ~MRENTHUSIASM | {
"answer": "075",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
[asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy] | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 1; i < 13; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy] Initially, both hands point to the number $12$. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move.
Let $N$ be the number of sequences of $144$ hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the $144$ movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by $1000$. | {
"answer": "608",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying
the real and imaginary part of $z$ are both integers;
$|z|=\sqrt{p},$ and
there exists a triangle whose three side lengths are $p,$ the real part of $z^{3},$ and the imaginary part of $z^{3}.$ | {
"answer": "349",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\text{eight}}.$ | {
"answer": "585",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be an isosceles triangle with $\angle A = 90^\circ.$ There exists a point $P$ inside $\triangle ABC$ such that $\angle PAB = \angle PBC = \angle PCA$ and $AP = 10.$ Find the area of $\triangle ABC.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(200); pair A, B, C, P; A = origin; B = (0,10*sqrt(5)); C = (10*sqrt(5),0); P = intersectionpoints(Circle(A,10),Circle(C,20))[0]; dot("$A$",A,1.5*SW,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*SE,linewidth(4)); dot("$P$",P,1.5*NE,linewidth(4)); markscalefactor=0.125; draw(rightanglemark(B,A,C,10),red); draw(anglemark(P,A,B,25),red); draw(anglemark(P,B,C,25),red); draw(anglemark(P,C,A,25),red); add(pathticks(anglemark(P,A,B,25), n = 1, r = 0.1, s = 10, red)); add(pathticks(anglemark(P,B,C,25), n = 1, r = 0.1, s = 10, red)); add(pathticks(anglemark(P,C,A,25), n = 1, r = 0.1, s = 10, red)); draw(A--B--C--cycle^^P--A^^P--B^^P--C); label("$10$",midpoint(A--P),dir(-30),blue); [/asy] ~MRENTHUSIASM | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x,y,$ and $z$ be real numbers satisfying the system of equations \begin{align*} xy + 4z &= 60 \\ yz + 4x &= 60 \\ zx + 4y &= 60. \end{align*} Let $S$ be the set of possible values of $x.$ Find the sum of the squares of the elements of $S.$ | {
"answer": "273",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | {
"answer": "719",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] unitsize(2cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0),dashed); [/asy] | {
"answer": "035",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle. | {
"answer": "928",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\] | {
"answer": "024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y,$ respectively. Suppose $PX=10,$ $PY=14,$ and $PQ=5.$ Then the area of trapezoid $XABY$ is $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n.$ | {
"answer": "033",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ \[\begin{array}{|c|c|c|c|c|c|} \hline \,1\, & \,3\, & \,5\, & \,7\, & \,9\, & 11 \\ \hline \,2\, & \,4\, & \,6\, & \,8\, & 10 & 12 \\ \hline \end{array}\] | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of collections of $16$ distinct subsets of $\{1,2,3,4,5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X \cap Y \not= \emptyset.$ | {
"answer": "081",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$ with side lengths $AB = 13,$ $BC = 14,$ and $CA = 15,$ let $M$ be the midpoint of $\overline{BC}.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}.$ There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ.$ Then $AQ$ can be written as $\frac{m}{\sqrt{n}},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | {
"answer": "247",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$ | {
"answer": "167",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangle $ABDC$ is perpendicular to $\mathcal{P},$ vertex $B$ is $2$ meters above $\mathcal{P},$ vertex $C$ is $8$ meters above $\mathcal{P},$ and vertex $D$ is $10$ meters above $\mathcal{P}.$ The cube contains water whose surface is parallel to $\mathcal{P}$ at a height of $7$ meters above $\mathcal{P}.$ The volume of water is $\frac{m}{n}$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Diagram
[asy] //Made by Djmathman size(250); defaultpen(linewidth(0.6)); pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); draw(A--B--Z--X--A--Y--C--X^^C--D--Z); draw(P1--P2--P3--P4--cycle^^D--P4); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); label("$\mathcal P$",(-13,4.5)); [/asy] | {
"answer": "751",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$ | {
"answer": "363",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Observe the following equations:
1. $\cos 2\alpha = 2\cos^2\alpha - 1$;
2. $\cos 4\alpha = 8\cos^4\alpha - 8\cos^2\alpha + 1$;
3. $\cos 6\alpha = 32\cos^6\alpha - 48\cos^4\alpha + 18\cos^2\alpha - 1$;
4. $\cos 8\alpha = 128\cos^8\alpha - 256\cos^6\alpha + 160\cos^4\alpha - 32\cos^2\alpha + 1$;
5. $\cos 10\alpha = m\cos^{10}\alpha - 1280\cos^8\alpha + 1120\cos^6\alpha + n\cos^4\alpha + p\cos^2\alpha - 1$.
It can be inferred that $m - n + p =$ ______. | {
"answer": "962",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Leah and Jackson run for 45 minutes on a circular track. Leah runs clockwise at 200 m/min in a lane with a radius of 40 meters, while Jackson runs counterclockwise at 280 m/min in a lane with a radius of 55 meters, starting on the same radial line as Leah. Calculate how many times they pass each other after the start. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the domains of the functions \( f(x) \) and \( g(x) \) are both the set of non-negative real numbers, and for any \( x \geq 0 \), \( f(x) \cdot g(x) = \min \{ f(x), g(x) \} \), given \( f(x) = 3 - x \) and \( g(x) = \sqrt{2x + 5} \), then the maximum value of \( f(x) \cdot g(x) \) is ______ . | {
"answer": "2\\sqrt{3} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $\angle B=30^{\circ}$, the area of $\triangle ABC$ is $\frac{3}{2}$, and $\sin A + \sin C = 2\sin B$. Calculate the value of $b$. | {
"answer": "\\sqrt{3}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | {
"answer": "10\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tournament with 2017 participating teams, each round consists of three randomly chosen teams competing, with exactly one team surviving from each round. If only two teams remain, a one-on-one battle determines the winner. How many battles must take place to declare a champion? | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the coordinates of a point on the terminal side of angle $\alpha$ are $(\frac{\sqrt{3}}{2},-\frac{1}{2})$, determine the smallest positive value of angle $\alpha$. | {
"answer": "\\frac{11\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Christina draws a pair of concentric circles. She draws chords $\overline{DE}$, $\overline{EF}, \ldots$ of the larger circle, each chord being tangent to the smaller circle. If $m\angle DEF = 85^\circ$, how many segments will she draw before returning to her starting point at $D$? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $a\cos B=(3c-b)\cos A$.
$(1)$ Find the value of $\cos A$;
$(2)$ If $b=3$, and point $M$ is on the line segment $BC$, $\overrightarrow{AB} + \overrightarrow{AC} = 2\overrightarrow{AM}$, $|\overrightarrow{AM}| = 3\sqrt{2}$, find the area of $\triangle ABC$. | {
"answer": "7\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\left(\frac{{a}^{2}-1}{a-3}-a-1\right) \div \frac{a+1}{{a}^{2}-6a+9}$, where $a=3-\sqrt{2}$. | {
"answer": "-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent? | {
"answer": "7/99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $2\sin 2A+\sin (A-B)=\sin C$, and $A\neq \frac{\pi}{2}$.
- (I) Find the value of $\frac{a}{b}$;
- (II) If $c=2$ and $C= \frac{\pi}{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{2 \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Angry reviews about the work of an online store are left by $80\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using this statistic, estimate the probability that the next customer will be satisfied with the service in this online store. | {
"answer": "0.64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area of a triangle with side lengths 7, 7, and 5. | {
"answer": "\\frac{5\\sqrt{42.75}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that positive integers \( a, b, c \) (\( a < b < c \)) form a geometric sequence, and
\[ \log_{2016} a + \log_{2016} b + \log_{2016} c = 3, \]
find the maximum value of \( a + b + c \). | {
"answer": "4066273",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
31 cars simultaneously started from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another on a lap, they collide, both go off the track, and are out of the race. In the end, one car remains. At what speed is it traveling? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given $a=4$, $b=5$, and the area of $\triangle ABC$ is $5\sqrt{3}$, find the values of $c$ and $\sin A$. | {
"answer": "\\frac{2\\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle centered at $O$, point $A$ is on the circle, and $\overline{BA}$ is tangent to the circle at $A$. Triangle $ABC$ is right-angled at $A$ with $\angle ABC = 45^\circ$. The circle intersects $\overline{BO}$ at $D$. Chord $\overline{BC}$ also extends to meet the circle at another point, $E$. What is the value of $\frac{BD}{BO}$?
A) $\frac{3 - \sqrt{3}}{2}$
B) $\frac{\sqrt{3}}{2}$
C) $\frac{2 - \sqrt{2}}{2}$
D) $\frac{1}{2}$
E) $\frac{\sqrt{2}}{2}$ | {
"answer": "\\frac{2 - \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three faces of a rectangular box meet at a corner, and the centers of these faces form the vertices of a triangle with side lengths of 4 cm, 5 cm, and 6 cm. What is the volume of the box, in cm^3? | {
"answer": "90 \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain exam, 6 questions are randomly selected from 20 questions. If a student can correctly answer at least 4 of these questions, they pass the exam. If they can correctly answer at least 5 of these questions, they achieve an excellent grade. It is known that a certain student can correctly answer 10 of these questions and that they have already passed the exam. The probability that they achieve an excellent grade is \_\_\_\_\_\_. | {
"answer": "\\frac{13}{58}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the obtuse angle $\alpha$ that satisfies the equation $$\frac {sin\alpha-3cos\alpha}{cos\alpha -sin\alpha }=tan2\alpha$$, find the value of $tan\alpha$. | {
"answer": "2 - \\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$ as $$\begin{cases} x=t \\ y= \frac { \sqrt {2}}{2}+ \sqrt {3}t \end{cases}$$ (where $t$ is the parameter), if the origin $O$ of the Cartesian coordinate system $xOy$ is taken as the pole and the direction of $Ox$ as the polar axis, and the same unit of length is chosen to establish the polar coordinate system, then the polar equation of curve $C$ is $\rho=2\cos\left(\theta- \frac {\pi}{4}\right)$.
(1) Find the angle of inclination of line $l$ and the Cartesian equation of curve $C$;
(2) If line $l$ intersects curve $C$ at points $A$ and $B$, and let point $P(0, \frac { \sqrt {2}}{2})$, find $|PA|+|PB|$. | {
"answer": "\\frac { \\sqrt {10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) with its right focus at $F(c, 0)$. A circle centered at the origin $O$ with radius $c$ intersects the hyperbola in the first quadrant at point $A$. The tangent to the circle at point $A$ has a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers of the form $\overline{a b c d a b c d}$ are divisible by 18769? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the square of a number $y^2$ is the sum of squares of 11 consecutive integers, find the minimum value of $y^2$. | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ have an eccentricity $e = \frac{1}{2}$, and the distance from the right focus to the line $\frac{x}{a} + \frac{y}{b} = 1$ be $d = \frac{\sqrt{21}}{7}$. Let $O$ be the origin.
$(1)$ Find the equation of the ellipse $E$;
$(2)$ Draw two perpendicular rays from point $O$ that intersect the ellipse $E$ at points $A$ and $B$, respectively. Find the distance from point $O$ to the line $AB$. | {
"answer": "\\frac{2\\sqrt{21}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression: $\frac{8}{1+a^{8}} + \frac{4}{1+a^{4}} + \frac{2}{1+a^{2}} + \frac{1}{1+a} + \frac{1}{1-a}$ and find its value when $a=2^{-\frac{1}{16}}$. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elliot and Emily run a 12 km race. They start at the same point, run 6 km up a hill, and return to the starting point by the same route. Elliot has a 8 minute head start and runs at the rate of 12 km/hr uphill and 18 km/hr downhill. Emily runs 14 km/hr uphill and 20 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)?
A) $\frac{161}{48}$
B) $\frac{169}{48}$
C) $\frac{173}{48}$
D) $\frac{185}{48}$ | {
"answer": "\\frac{169}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a \in \{0, 1, 2\}$ and $b \in \{-1, 1, 3, 5\}$, find the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function on the interval $(1, +\infty)$. | {
"answer": "\\frac{5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\[
\frac{\left(\left(4.625 - \frac{13}{18} \cdot \frac{9}{26}\right) : \frac{9}{4} + 2.5 : 1.25 : 6.75\right) : 1 \frac{53}{68}}{\left(\frac{1}{2} - 0.375\right) : 0.125 + \left(\frac{5}{6} - \frac{7}{12}\right) : (0.358 - 1.4796 : 13.7)}
\]
| {
"answer": "\\frac{17}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice has six magical pies in her pocket - two that make you grow and the rest make you shrink. When Alice met Mary Ann, she blindly took three pies out of her pocket and gave them to Mary. Find the probability that one of the girls has no growth pies. | {
"answer": "0.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest exact square with last digit not $0$ , such that after deleting its last two digits we shall obtain another exact square. | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression
$$
\sqrt{x^{2}-2 \sqrt{3} \cdot|x|+4}+\sqrt{x^{2}+2 \sqrt{3} \cdot|x|+12}
$$
as well as the values of $x$ at which it is achieved. | {
"answer": "2 \\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system \( xOy \), the equation of ellipse \( C \) is \( \frac{x^2}{9} + \frac{y^2}{10} = 1 \). Let \( F \) and \( A \) be the upper focus and the right vertex of ellipse \( C \), respectively. If \( P \) is a point on ellipse \( C \) located in the first quadrant, find the maximum value of the area of quadrilateral \( OAPF \). | {
"answer": "\\frac{3\\sqrt{11}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sides of triangle $\triangle ABC$ opposite to angles $A$, $B$, and $C$ are in arithmetic sequence and $C=2\left(A+B\right)$, calculate the ratio $\frac{b}{a}$. | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them? | {
"answer": "\\frac{\\sqrt{6} + 2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $$cos(α- \frac {π}{6})-sinα= \frac {2 \sqrt {3}}{5}$$, find the value of $$cos(α+ \frac {7π}{6})$$. | {
"answer": "- \\frac {2 \\sqrt {3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Numbers between $200$ and $500$ that are divisible by $5$ contain the digit $3$. How many such whole numbers exist? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many cubic centimeters are in the volume of a cone having a diameter of 12 cm and a slant height of 10 cm? | {
"answer": "96 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya and Vasya took a math test. Petya answered $80\%$ of all the questions correctly, while Vasya answered exactly 35 questions correctly. The number of questions both answered correctly is exactly half the total number of questions. No one answered 7 questions. How many questions were on the test? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A swimming pool is in the shape of a circle with diameter 60 ft. The depth varies linearly along the east-west direction from 3 ft at the shallow end in the east to 15 ft at the diving end in the west but does not vary at all along the north-south direction. What is the volume of the pool, in cubic feet (ft³)? | {
"answer": "8100 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During an underwater archaeological activity, a diver needs to dive $50$ meters to the bottom of the water for archaeological work. The oxygen consumption consists of the following three aspects: $(1)$ The average diving speed is $x$ meters/minute, and the oxygen consumption per minute is $\frac{1}{100}x^{2}$ liters; $(2)$ The working time at the bottom of the water is between $10$ and $20$ minutes, and the oxygen consumption per minute is $0.3$ liters; $(3)$ When returning to the water surface, the average speed is $\frac{1}{2}x$ meters/minute, and the oxygen consumption per minute is $0.32$ liters. The total oxygen consumption of the diver in this archaeological activity is $y$ liters.
$(1)$ If the working time at the bottom of the water is $10$ minutes, express $y$ as a function of $x$;
$(2)$ If $x \in [6, 10]$, the working time at the bottom of the water is $20$ minutes, find the range of total oxygen consumption $y$;
$(3)$ If the diver carries $13.5$ liters of oxygen, how many minutes can the diver stay underwater at most (round to the nearest whole number)? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum $C_{3}^{2}+C_{4}^{2}+C_{5}^{2}+\ldots+C_{19}^{2}$. | {
"answer": "1139",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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