problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\frac{1}{2}$, what is its speed? | 3 \sqrt{3} - 3 | 0 | 8,042.375 | -1 | 8,042.375 | |
A clothing store buys 600 pairs of gloves at a price of 12 yuan per pair. They sell 470 pairs at a price of 14 yuan per pair, and the remaining gloves are sold at a price of 11 yuan per pair. What is the total profit made by the clothing store from selling this batch of gloves? | 810 | 0.4375 | 620.1875 | 641.142857 | 603.888889 | |
Find the shortest distance from a point on the curve $y=x^{2}-\ln x$ to the line $x-y-2=0$. | \sqrt{2} | 0.8125 | 5,656.1875 | 5,071 | 8,192 | |
Given an ellipse C centered at the origin with its left focus F($-\sqrt{3}$, 0) and right vertex A(2, 0).
(1) Find the standard equation of ellipse C;
(2) A line l with a slope of $\frac{1}{2}$ intersects ellipse C at points A and B. Find the maximum value of the chord length |AB| and the equation of line l at this ... | \sqrt{10} | 0 | 7,339 | -1 | 7,339 | |
The sequence $\left\{a_{n}\right\}$ satisfies $a_{1} = 1$, and for each $n \in \mathbf{N}^{*}$, $a_{n}$ and $a_{n+1}$ are the roots of the equation $x^{2} + 3n x + b_{n} = 0$. Find $\sum_{k=1}^{20} b_{k}$. | 6385 | 0.0625 | 8,114.375 | 6,950 | 8,192 | |
Calculate the value of the following product as a common fraction:
\[ \left(2 \cdot \left(1-\frac{1}{2}\right)\right) \cdot \left(1-\frac{1}{3}\right) \cdot \left(1-\frac{1}{4}\right) \dotsm \left(1-\frac{1}{50}\right) \] | \frac{1}{50} | 0 | 5,348.25 | -1 | 5,348.25 | |
Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_. | 12\pi | 0.0625 | 8,066.8125 | 7,692 | 8,091.8 | |
The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990.$ Find the greatest number of apples ... | 220 | Let the terms in the sequence be defined as \[a_1, a_2, ..., a_6.\]
Since this is an arithmetic sequence, we have $a_1+a_6=a_2+a_5=a_3+a_4.$ So, \[\sum_{i=1}^6 a_i=3(a_1+a_6)=990.\] Hence, $(a_1+a_6)=330.$ And, since we are given that $a_6=2a_1,$ we get $3a_1=330\implies a_1=110$ and $a_6=\boxed{220}.$
~Kiran | 1 | 1,573.6875 | 1,573.6875 | -1 |
A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | 80 | 0 | 8,192 | -1 | 8,192 | |
If a polygon is formed by connecting a point on one side of the polygon to all its vertices, forming $2023$ triangles, determine the number of sides of this polygon. | 2024 | 0.3125 | 4,887.8125 | 3,801.4 | 5,381.636364 | |
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:
[b](i)[/b] the numbers $ x$, $ y$, $ z$... | n \in \{69, 84\} |
Let us consider a set \( S = \{1, 2, \ldots, n\} \) whose elements are to be colored either red or blue. We need to find all positive integers \( n \) for which the set \( S \times S \times S \) contains exactly 2007 ordered triples \( (x, y, z) \) satisfying the following conditions:
1. The numbers \( x \), \( y \), ... | 0 | 8,171.125 | -1 | 8,171.125 |
In the figure, $\overline{DC}$ is parallel to $\overline{AB}$. We have $\angle DCA = 40^\circ$ and $\angle ABC = 73^\circ$. Find $\angle ACB$.
[asy]
pair A,B,C,D;
A = (0,0);
B= (1,0);
D = (-0.1,0.6);
C = (0.8,0.6);
draw((-0.2,0)--(1.2,0));
draw((-0.2,0.6)--(1.2,0.6));
draw(A--C--B);
label("$A$",A,S);
label("$B$",B,S)... | 67^\circ | 0.8125 | 4,708.625 | 4,177.384615 | 7,010.666667 | |
Let \( a_{1}, a_{2}, \cdots, a_{k}\left(k \in \mathbf{Z}_{+}\right) \) be integers greater than 1, and they satisfy
\[
\left(a_{1}!\right)\left(a_{2}!\right) \cdots\left(a_{k}!\right) \mid 2017!
\]
Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) as \( k \) varies. | 5024 | 0 | 8,192 | -1 | 8,192 | |
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 $m + n$. | 47 | By Bretschneider's Formula, \[30=\tfrac{1}{4}\sqrt{4u^2v^2-(b^2+d^2-a^2-c^2)^2}=\tfrac{1}{4}\sqrt{4u^2v^2-441}.\] Thus, $uv=3\sqrt{1649}$. Also, \[[ABCD]=\tfrac 12 \cdot uv\sin{\theta};\] solving for $\sin{\theta}$ yields $\sin{\theta}=\tfrac{40}{\sqrt{1649}}$. Since $\theta$ is acute, $\cos{\theta}$ is positive, from ... | 0 | 8,071.9375 | -1 | 8,071.9375 |
How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not. | 22 | 0.0625 | 7,900.9375 | 6,857 | 7,970.533333 | |
Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \geq 5$. | 120 | 5!=120. | 0.125 | 7,076.4375 | 4,732 | 7,411.357143 |
Given two squares $ABCD$ and $DCFE$ with side lengths of $1$, where the planes they reside in are perpendicular to each other. Points $P$ and $Q$ are moving points on line segments $BC$ and $DE$ (including endpoints), with $PQ = \sqrt{2}$. Let the trajectory of the midpoint of line segment $PQ$ be curve $\mathcal{A}$. ... | \frac{\pi}{4} | 0.4375 | 6,765.375 | 5,509.571429 | 7,742.111111 | |
Given the ellipse $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the hyperbola $C\_2$: $x^{2}- \frac{y^{2}}{4}=1$ share a common focus. One of the asymptotes of $C\_2$ intersects with the circle having the major axis of $C\_1$ as its diameter at points $A$ and $B$. If $C\_1$ precisely trisects the... | \sqrt{2} | 0.875 | 6,987.875 | 6,815.857143 | 8,192 | |
A sequence consists of $3000$ terms. Each term after the first is $3$ larger than the previous term. The sum of the $3000$ terms is $12000$. Calculate the sum when every third term is added up, starting with the first term and ending with the third to last term. | 1000 | 0.3125 | 6,744.75 | 4,859 | 7,601.909091 | |
In the senior year of high school, the weights of 8 students are $90$, $100$, $110$, $120$, $140$, $150$, $150$, $160 (unit: kg)$. Now, if 3 students are selected from them to participate in a tug-of-war, the probability that the student with the highest weight among the selected students is exactly the $70$th percenti... | \frac{25}{56} | 0.1875 | 5,866.3125 | 6,629.333333 | 5,690.230769 | |
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$. | 384 | [asy]defaultpen(fontsize(9)); pair A=expi(-pi/2-acos(475/486)), B=expi(-pi/2+acos(475/486)), C=expi(-pi/2+acos(475/486)+acos(7/18)), D=expi(-pi/2+acos(475/486)+2*acos(7/18)), E=expi(-pi/2+acos(475/486)+3*acos(7/18)), F=expi(-pi/2-acos(475/486)-acos(7/18)); draw(unitcircle);draw(A--B--C--D--E--F--A);draw(A--C..A--D..A--... | 0.125 | 8,129.5625 | 7,791 | 8,177.928571 |
In parallelogram $ABCD$, angle $B$ measures $110^\circ$. What is the number of degrees in the measure of angle $C$? | 70^\circ | 1 | 1,084.8125 | 1,084.8125 | -1 | |
Professor Ma has formulated n different but equivalent statements A_{1}, A_{2}, \ldots, A_{n}. Every semester, he advises a student to prove an implication A_{i} \Rightarrow A_{j}, i \neq j. This is the dissertation topic of this student. Every semester, he has only one student, and we assume that this student finishes... | \[
\frac{1}{2}(n+2)(n-1)
\] | We will first construct an answer with \frac{1}{2}(n+2)(n-1) students. Then, we will show this is the best possible answer. Construction: First, (n-1) students sequentially prove A_{1} \Rightarrow A_{i} for i=2, \ldots, n. Then, (n-2) students sequentially prove A_{2} \Rightarrow A_{i} for i=3, \ldots, n. Continue this... | 0 | 8,098.875 | -1 | 8,098.875 |
Given the line y=b intersects with the function f(x)=2x+3 and the function g(x)=ax+ln x (where a is a real constant in the interval [0, 3/2]), find the minimum value of |AB|. | 2 - \frac{\ln 2}{2} | 0.125 | 7,974.625 | 6,514 | 8,183.285714 | |
Given that $\binom{20}{13} = 77520$, $\binom{20}{14} = 38760$ and $\binom{18}{12} = 18564$, find $\binom{19}{13}$. | 27132 | 0.625 | 5,355.1875 | 3,653.1 | 8,192 | |
Given the functions $f(x)=x^{2}+ax+3$, $g(x)=(6+a)\cdot 2^{x-1}$.
(I) If $f(1)=f(3)$, find the value of the real number $a$;
(II) Under the condition of (I), determine the monotonicity of the function $F(x)=\frac{2}{1+g(x)}$ and provide a proof;
(III) When $x \in [-2,2]$, $f(x) \geqslant a$, ($a \notin (-4,4)$) alwa... | -7 | 0.5 | 5,968.1875 | 5,194.375 | 6,742 | |
A circle with radius $r$ is tangent to sides $AB, AD$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. The area of the rectangle, in terms of $r$, is | $8r^2$ | Let's analyze the problem and derive the solution step-by-step.
1. **Understanding the Geometry**:
- The circle is tangent to sides $AB$, $AD$, and $CD$ of rectangle $ABCD$. This implies that the radius $r$ of the circle is the distance from the center of the circle to each of these three sides.
- The circle pas... | 0 | 3,923.9375 | -1 | 3,923.9375 |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | 40 | 0.25 | 7,458.0625 | 6,014.25 | 7,939.333333 | |
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$. | 720 | Begin as in solution 2, drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Apply law of cosines on $\theta = {30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}}$ to get $d^2 = 288 - 288 \cos \theta$ where $d$ is the diagonal or sidelength distance between two p... | 0.25 | 7,252.3125 | 5,924 | 7,695.083333 |
Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns. Phil buys a number of these packs for a barbecue. After the barbecue, Phil finds that he has 4 hot dogs left over. What is the SECOND smallest number of packs of hot dogs he could have bought? | 6 | 0.875 | 3,499.125 | 2,828.714286 | 8,192 | |
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = 2g(x) + g(g(y) - g(-x)) + y \] for all real numbers \( x \) and \( y \).
Let \( n \) be the number of possible values of \( g(2) \), and let \( s \) be the sum of all possible values of \( g(2) \). Find \( n \times s \). | -2 | 0 | 8,192 | -1 | 8,192 | |
A string has been cut into 4 pieces, all of different lengths. The length of each piece is 2 times the length of the next smaller piece. What fraction of the original string is the longest piece? | \frac{8}{15} | Let $L$ be the length of the string. If $x$ is the length of the shortest piece, then since each of the other pieces is twice the length of the next smaller piece, then the lengths of the remaining pieces are $2x, 4x$, and $8x$. Since these four pieces make up the full length of the string, then $x+2x+4x+8x=L$ or $15x=... | 1 | 2,656.9375 | 2,656.9375 | -1 |
Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\$90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price... | 750 |
Let the sticker price of the computer be denoted as \( x \).
**Store A Calculation:**
- Store A offers a 15% discount, so the price after the discount is \( 0.85x \).
- After applying the additional $90 rebate, the final price at store A becomes \( 0.85x - 90 \).
**Store B Calculation:**
- Store B offers a 25% disco... | 1 | 2,219.4375 | 2,219.4375 | -1 |
All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{3} | 0.5 | 7,298.8125 | 6,405.625 | 8,192 | |
Given the function $f(x) = \frac{1}{3}x^3 - 4x + 4$,
(I) Find the extreme values of the function;
(II) Find the maximum and minimum values of the function on the interval [-3, 4]. | -\frac{4}{3} | 1 | 2,897.6875 | 2,897.6875 | -1 | |
How many ordered pairs $(m,n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$? | 4 | To find the number of ordered pairs $(m,n)$ of positive integers such that $m \ge n$ and $m^2 - n^2 = 96$, we start by factoring the expression for the difference of squares:
\[ m^2 - n^2 = (m+n)(m-n) = 96. \]
We need to find all pairs of factors of $96$ that can be expressed as $(m+n, m-n)$. Since $m$ and $n$ are in... | 0.9375 | 5,247.6875 | 5,051.4 | 8,192 |
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$. | \frac{1}{13} | 0 | 5,018.1875 | -1 | 5,018.1875 | |
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle? | \frac{1}{3} | 1. **Identify the Problem Setup**: We have two concentric circles with radii $1$ and $2$. We need to find the probability that a randomly chosen chord on the outer circle intersects the inner circle.
2. **Choosing a Reference Point**: Let $O$ be the center of both circles. Choose a point $A$ on the outer circle arbitr... | 0.5 | 7,500.125 | 6,808.25 | 8,192 |
In two places, A and B, the monthly telephone rental fee is $18. The monthly telephone fee $y (in dollars)$ varies with time $x (in minutes)$. Based on the data in the table below, answer the following questions:
| Talk Time $x (minutes)$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
|-------------------------|-----|-----|---... | 20 | 0.8125 | 1,387.375 | 1,449.153846 | 1,119.666667 | |
Quadrilateral $ABCD$ is a parallelogram with an area of $50$ square units. Points $P$ and $Q$ are located on sides $AB$ and $CD$ respectively, such that $AP = \frac{1}{3}AB$ and $CQ = \frac{2}{3}CD$. What is the area of triangle $APD$? | 16.67 | 0 | 4,569.6875 | -1 | 4,569.6875 | |
Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$? | 1009 | 0 | 7,916.125 | -1 | 7,916.125 | |
The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. | 306 | 0.125 | 7,943.125 | 6,621.5 | 8,131.928571 | |
In a spelling bee $50\%$ of the students were eliminated after the first round. Only $\frac{1}{3}$ of the remaining students were still in the contest after the second round. If 24 students were still in the contest after the second round, how many students began the contest? | 144 | 1 | 1,190.125 | 1,190.125 | -1 | |
Compute $1-2+3-4+\dots+100-101$. | 51 | 0.875 | 4,292.25 | 3,735.142857 | 8,192 | |
A sports league consists of 16 teams divided into 2 divisions of 8 teams each. Throughout the season, each team plays every other team in its own division three times and every team in the other division twice. How many games are in a complete season for the league? | 296 | 0.4375 | 6,737.5625 | 5,293.857143 | 7,860.444444 | |
Given the equation $x^2+kx+6=0$ has one root as $2$, find the other root and the value of $k$. | -5 | 1 | 1,972.0625 | 1,972.0625 | -1 | |
The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A,$ respectively. In this proc... | 392 | 0.0625 | 7,918.3125 | 7,574 | 7,941.266667 | |
Given the chord length intercepted by the circle $x^{2}+y^{2}+2x-4y+1=0$ on the line $ax-by+2=0$ $(a > 0, b > 0)$ is 4, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{3}{2} + \sqrt{2} | 0.0625 | 5,912.9375 | 5,864 | 5,916.2 | |
Given that the terminal side of angle $\alpha$ ($0 < \alpha < \frac{\pi}{2}$) passes through the point $(\cos 2\beta, 1+\sin 3\beta \cos \beta - \cos 3\beta \sin \beta)$, where $\frac{\pi}{2} < \beta < \pi$ and $\beta \neq \frac{3\pi}{4}$, calculate $\alpha - \beta$. | -\frac{3\pi}{4} | 0.0625 | 8,108.125 | 6,850 | 8,192 | |
$(1)$ Calculate: $tan60°×{({-2})^{-1}}-({\sqrt{\frac{3}{4}}-\sqrt[3]{8}})+|{-\frac{1}{2}\sqrt{12}}|$;<br/>$(2)$ Simplify and find the value: $({\frac{{x+2}}{{{x^2}-2x}}-\frac{{x-1}}{{{x^2}-4x+4}}})÷\frac{{x-4}}{x}$, where $x=\sqrt{2}+2$. | \frac{1}{2} | 1 | 3,753.3125 | 3,753.3125 | -1 | |
Given that a smaller circle is entirely inside a larger circle, such that the larger circle has a radius $R = 2$, and the areas of the two circles form an arithmetic progression, with the largest area being that of the larger circle, find the radius of the smaller circle. | \sqrt{2} | 0 | 8,042.6875 | -1 | 8,042.6875 | |
If $(2)(3)(4) = 6x$, what is the value of $x$? | 4 | Since $(2)(3)(4) = 6x$, then $6(4) = 6x$. Dividing both sides by 6, we obtain $x = 4$. | 1 | 956.1875 | 956.1875 | -1 |
Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-50^2).\] How many integers $n$ are there such that $P(n)\leq 0$? | 1300 | 0 | 8,159.5 | -1 | 8,159.5 | |
An $n \times m$ maze is an $n \times m$ grid in which each cell is one of two things: a wall, or a blank. A maze is solvable if there exists a sequence of adjacent blank cells from the top left cell to the bottom right cell going through no walls. (In particular, the top left and bottom right cells must both be blank.)... | 3 | We must have both top-left and bottom-right cells blank, and we cannot have both top-right and bottom-left cells with walls. As long as those conditions are satisfied, the maze is solvable, so the answer is 3. | 0.3125 | 6,897.625 | 5,135.4 | 7,698.636364 |
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.
Diagram
Let... | 112 | Either $PA=PC$ or $PD=PB$. Let $PD=PB=s$. Applying Stewart's Theorem on $\triangle ABD$ and $\triangle BCD$, dividing by $2s$ and rearranging, \[\tag{1}CP^2+s^2=148\] \[\tag{2}AP^2+s^2=180\] Applying Stewart on $\triangle CAB$ and $\triangle CAD$, \[\tag{3} 5CP^2=3AP^2\] Substituting equations 1 and 2 into 3 and rearra... | 0 | 8,192 | -1 | 8,192 |
Completely factor the following expression: $9x^2+3x$. | 3x(3x+1) | 0.9375 | 1,429.875 | 1,454.533333 | 1,060 | |
Find the area of trapezoid \(ABCD\) with a side \(BC = 5\), where the distances from vertices \(A\) and \(D\) to the line \(BC\) are 3 and 7 respectively. | 25 | 0.1875 | 8,081.375 | 8,044.666667 | 8,089.846154 | |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of line segment \( BM \). Given that \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \). | \sqrt{3} + 1 | 0 | 6,952.1875 | -1 | 6,952.1875 | |
Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $150$. | 100 | 0 | 8,192 | -1 | 8,192 | |
If \( 3-5+7=6-x \), what is the value of \( x \)? | 1 | Simplifying the left side of the equation, we obtain \( 5=6-x \). Therefore, \( x=6-5=1 \). | 1 | 1,406.1875 | 1,406.1875 | -1 |
What is $\frac{1357_{9}}{100_{4}}-2460_{8}+5678_{9}$? Express your answer in base 10. | 2938 | 1 | 3,082.875 | 3,082.875 | -1 | |
Find all negative values of \( a \) for which the equation
$$
\frac{8 \pi a - \arcsin (\sin x) + 3 \arccos (\cos x) - a x}{3 + \operatorname{tg}^{2} x} = 0
$$
has exactly three solutions. Provide the sum of all found \( a \) (if no such \( a \) exist, indicate 0; if the sum of \( a \) is not an integer, round it to t... | -2.47 | 0 | 8,192 | -1 | 8,192 | |
Two distinct numbers a and b are chosen randomly from the set $\{3, 3^2, 3^3, ..., 3^{15}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
A) $\frac{1}{10}$
B) $\frac{2}{7}$
C) $\frac{1}{7}$
D) $\frac{1}{5}$ | \frac{2}{7} | 0 | 6,606.3125 | -1 | 6,606.3125 | |
Given the sequence $\{a_n\}$ satisfies $\{a_1=2, a_2=1,\}$ and $\frac{a_n \cdot a_{n-1}}{a_{n-1}-a_n}=\frac{a_n \cdot a_{n+1}}{a_n-a_{n+1}}(n\geqslant 2)$, determine the $100^{\text{th}}$ term of the sequence $\{a_n\}$. | \frac{1}{50} | 0.6875 | 5,810.3125 | 5,020.363636 | 7,548.2 | |
Let \[P(x) = (3x^4 - 39x^3 + ax^2 + bx + c)(4x^4 - 96x^3 + dx^2 + ex + f),\] where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\{1, 2, 2, 3, 3, 4, 6\}.$ Find $P(7).$ | 86400 | 0 | 7,489.875 | -1 | 7,489.875 | |
How many multiples of 4 are between 100 and 350? | 62 | 0.8125 | 4,416 | 3,861.076923 | 6,820.666667 | |
All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? | 70 | To solve this problem, we need to count the number of distinct intersection points formed by the diagonals of a regular octagon. We will use combinatorial geometry to determine this count.
#### Step 1: Counting the Total Number of Diagonals
In an $n$-sided polygon, the number of diagonals can be calculated using the f... | 0.125 | 7,838.625 | 5,365 | 8,192 |
Given an acute angle \( \theta \), the equation \( x^{2} + 4x \cos \theta + \cot \theta = 0 \) has a double root. Find the radian measure of \( \theta \). | \frac{5\pi}{12} | 0.4375 | 7,360.5 | 7,235.857143 | 7,457.444444 | |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | 31 | 1 | 2,535.875 | 2,535.875 | -1 | |
Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's... | 25 | 1. **Define the number of comic books over time for both Kymbrea and LaShawn:**
- Kymbrea's collection: Initially, she has 30 comic books and adds 2 each month. Thus, after $x$ months, she will have:
\[
30 + 2x
\]
- LaShawn's collection: Initially, he has 10 comic books and adds 6 each month. Thus,... | 1 | 2,778 | 2,778 | -1 |
Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers \underline{a} \underline{b} \underline{c}, if \underline{a} \underline{b} \underline{c} is a multiple of $x$, then the three-digit (base 10) number \underline{b} \underline{c} \underline{a} is also a multipl... | 64 | Note that $\overline{a b c 0}-\overline{b c a}=a\left(10^{4}-1\right)$ must also be a multiple of $x$. Choosing $a=1$ means that $x$ divides $10^{3}-1$, and this is clearly a necessary and sufficient condition. The only two-digit factors of $10^{3}-1$ are 27 and 37, so our answer is $27+37=64$. | 0 | 8,192 | -1 | 8,192 |
The diagonals of a trapezoid are perpendicular to each other, its height is 4 units, and one of its diagonals is 5 units. What is the area of the trapezoid? | 50/3 | 0.5625 | 7,005.375 | 6,312.111111 | 7,896.714286 | |
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals | 50^\circ | 1. **Identify the properties of triangle $ABC$**: Given that $AB = AC$, triangle $ABC$ is isosceles. Also, $\angle A = 80^\circ$. Since the sum of angles in a triangle is $180^\circ$, and $AB = AC$, the base angles $\angle B$ and $\angle C$ are equal. Thus, we calculate:
\[
\angle B = \angle C = \frac{180^\circ -... | 0.1875 | 8,153.75 | 7,988 | 8,192 |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | 6 | 1 | 2,154.4375 | 2,154.4375 | -1 | |
If \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) satisfy the following system of equations:
\[
\begin{cases}
2x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 6, \\
x_{1} + 2x_{2} + x_{3} + x_{4} + x_{5} = 12, \\
x_{1} + x_{2} + 2x_{3} + x_{4} + x_{5} = 24, \\
x_{1} + x_{2} + x_{3} + 2x_{4} + x_{5} = 48, \\
x_{1} + x_{2} + x_{3} + x_{... | 181 | 0.3125 | 6,766.1875 | 3,629.4 | 8,192 | |
The maximum value and the minimum positive period of the function $f(x)=\cos 4x \cdot \cos 2x \cdot \cos x \cdot \sin x$ are to be determined. | \frac{\pi}{4} | 0.625 | 5,717.875 | 4,233.4 | 8,192 | |
A box contains $3$ pennies, $5$ nickels, $7$ dimes, and $4$ quarters. Eight coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $1.00$ (100 cents)?
A) $\frac{325}{75582}$
B) $0$
C) $\frac{5000}{75582}$
D) $\fr... | \frac{2345}{75582} | 0 | 8,192 | -1 | 8,192 | |
Select 3 numbers from the range 1 to 300 such that their sum is exactly divisible by 3. How many such combinations are possible? | 1485100 | 0.1875 | 7,781.5 | 6,002.666667 | 8,192 | |
A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct barcodes are possible? | 116 | 0.0625 | 7,853.0625 | 8,192 | 7,830.466667 | |
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3,... | D | 1. **Calculate the total sum of the numbers on the slips**:
The numbers on the slips are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. Their total sum is:
\[
2+2+2+2.5+2.5+3+3+3+3+3.5+4+4.5 = 35
\]
2. **Determine the sums for each cup**:
Since Tom wants the sums in the cups to be consecutive integ... | 0 | 8,192 | -1 | 8,192 |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, if $\cos B= \frac {4}{5}$, $a=5$, and the area of $\triangle ABC$ is $12$, find the value of $\frac {a+c}{\sin A+\sin C}$. | \frac {25}{3} | 1 | 3,595.25 | 3,595.25 | -1 | |
In the interval $[0, \pi]$, a number $\vartheta$ is randomly selected. The probability that $\sqrt{2} \leq \sqrt{2}\cos\vartheta + \sqrt{2}\sin\vartheta \leq 2$ holds true is $\_\_\_\_\_\_\_\_\cdot$ | \frac{1}{2} | 0.4375 | 6,969 | 5,837.142857 | 7,849.333333 | |
Altitudes $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at $H$. If $\angle BAC = 46^\circ$ and $\angle ABC = 71^\circ$, then what is $\angle AHB$?
[asy]
size(150); defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (1.8,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
... | 117^\circ | 0.9375 | 4,204.3125 | 3,938.466667 | 8,192 | |
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Find the probability that
\[\sqrt{2+\sqrt{3}}\le\left|v+w\right|.\] | \frac{83}{499} | 0.3125 | 7,494.75 | 6,883.2 | 7,772.727273 | |
Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs. | 1680 | 0.125 | 6,575.6875 | 5,078 | 6,789.642857 | |
Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches.
Then the area of circle $II$, in square inches, is: | 16 | 1. **Understanding the Relationship Between the Circles**:
Given that Circle $I$ is tangent to Circle $II$ and passes through the center of Circle $II$, we can deduce that the diameter of Circle $I$ is equal to the radius of Circle $II$. This is because the point of tangency to the center of Circle $II$ implies that... | 1 | 2,342.5 | 2,342.5 | -1 |
A point $P$ is randomly placed in the interior of right triangle $ABC$, with coordinates $A(0,6)$, $B(9,0)$, and $C(0,0)$. What is the probability that the area of triangle $APC$ is more than a third of the area of triangle $ABC$? | \frac{1}{3} | 0 | 5,737.125 | -1 | 5,737.125 | |
Points are drawn on the sides of a square, dividing each side into \( n \) equal parts. The points are joined to form several small squares and some triangles. How many small squares are formed when \( n=7 \)? | 84 | 0 | 5,450.625 | -1 | 5,450.625 | |
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth. | 8.64 | 1 | 3,165.625 | 3,165.625 | -1 | |
If $\tan x = 2,$ then find $\tan \left( x + \frac{\pi}{4} \right).$ | -3 | 1 | 2,340.875 | 2,340.875 | -1 | |
Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse?
A) 35
B) 40
C) 50
D) 65
E) 95 | 40 | 0 | 5,811.1875 | -1 | 5,811.1875 | |
A rectangular chocolate bar is made of equal squares. Irena breaks off two complete strips of squares and eats the 12 squares she obtains. Later, Jack breaks off one complete strip of squares from the same bar and eats the 9 squares he obtains. How many squares of chocolate are left in the bar?
A) 72
B) 63
C) 54
D) 45
... | 45 | 0 | 7,986.0625 | -1 | 7,986.0625 | |
Compute the value of $$64^{-\frac{1}{3}}+lg0.001$$. | - \frac{11}{2} | 0 | 1,847.75 | -1 | 1,847.75 | |
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? | \frac{147}{1024} | To solve this problem, we need to consider the different ways two cubes can be painted such that they are identical in appearance after possible rotations. We will calculate the number of ways to paint each cube and then find the probability that they match.
#### Step 1: Count the total number of ways to paint the cub... | 0 | 7,967.5625 | -1 | 7,967.5625 |
Given vectors $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $|\overrightarrow{c}|=\sqrt{2}$, and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$, calculate $\cos 〈\overrightarrow{a}-\overrightarrow{c}$,$\overrightarrow{b}-\overrightarrow{c}〉$. | \frac{4}{5} | 0.9375 | 3,895.125 | 3,894 | 3,912 | |
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\text{o}$. The block is then sliced in half al... | 53 | 0 | 8,160.5625 | -1 | 8,160.5625 | |
Let $a, b, c, x, y$, and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, \quad b=\frac{c+a}{y-2}, \quad c=\frac{a+b}{z-2}$. If $x y+y z+z x=67$ and $x+y+z=2010$, find the value of $x y z$. | -5892 | Manipulate the equations to get a common denominator: $a=\frac{b+c}{x-2} \Longrightarrow x-2=$ $\frac{b+c}{a} \Longrightarrow x-1=\frac{a+b+c}{a} \Longrightarrow \frac{1}{x-1}=\frac{a}{a+b+c}$; similarly, $\frac{1}{y-1}=\frac{b}{a+b+c}$ and $\frac{1}{z-1}=\frac{c}{a+b+c}$. Thus $\frac{1}{x-1}+\frac{1}{y-1}+\frac{1}{z-1... | 0.25 | 6,750.0625 | 4,212.5 | 7,595.916667 |
Compute $\tan\left(\frac{\pi}{9}\right)\tan\left(\frac{2\pi}{9}\right)\tan\left(\frac{4\pi}{9}\right)$. | \frac{1}{3} | 0 | 7,518.5 | -1 | 7,518.5 | |
Given a parabola $C:y^2=2px (p > 0)$, the sum of the distances from any point $Q$ on the parabola to a point inside it, $P(3,1)$, and the focus $F$, has a minimum value of $4$.
(I) Find the equation of the parabola;
(II) Through the focus $F$, draw a line $l$ that intersects the parabola $C$ at points $A$ and $B$. Fi... | -3 | 0 | 8,192 | -1 | 8,192 | |
In right triangle \( \triangle ABC \) where \(\angle ACB = 90^\circ\), \(CA = 3\), and \(CB = 4\), there is a point \(P\) inside \(\triangle ABC\) such that the sum of the distances from \(P\) to the three sides is \(\frac{13}{5}\). Find the length of the locus of point \(P\). | \frac{\sqrt{5}}{2} | 0 | 4,732.8125 | -1 | 4,732.8125 |
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