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 |
|---|---|---|---|---|---|---|
Triangle $DEF$ is isosceles with angle $E$ congruent to angle $F$. The measure of angle $F$ is three times the measure of angle $D$. What is the number of degrees in the measure of angle $E$? | \frac{540}{7} | 0.9375 | 2,510.125 | 2,487.733333 | 2,846 | |
Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\triangle ABC$. If the length of segment $AX$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive inte... | 6710 | Note that the incircle of $\triangle ABC$ is the $A$-excenter of $\triangle AXY$. Let $r$ be the radius of this circle. We can compute the area of $\triangle AXY$ in two ways: $$\begin{aligned} K_{AXY} & =\frac{1}{2} \cdot AX \cdot AY \sin A \\ & =r \cdot(AX+AY-XY) / 2 \\ \Longrightarrow AY & =\frac{r}{\sin A} \end{ali... | 0.125 | 8,192 | 8,192 | 8,192 |
Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails? | 396 | After testing various values of $m$ in $f(m)$ of solution 1 to determine $m$ for which $f(m) = 1992$, we find that $m \in \{7980, 7981, 7982, 7983, 7984\}$. WLOG, we select $7980$. Furthermore, note that every time $k$ reaches a multiple of $25$, $k!$ will gain two or more additional factors of $5$ and will thus skip o... | 0 | 8,192 | -1 | 8,192 |
If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$? | 20 | 0 | 8,192 | -1 | 8,192 | |
If the probability that a baby born in a certain hospital will speak in the next day is 1/4, what is the probability that at least 2 babies out of a cluster of 5 babies will speak tomorrow? | \frac{47}{128} | 1 | 4,204.0625 | 4,204.0625 | -1 | |
Ten points are spaced around at equal intervals on the circumference of a regular pentagon, each side being further divided into two equal segments. Two of the 10 points are chosen at random. What is the probability that the two points are exactly one side of the pentagon apart?
A) $\frac{1}{5}$
B) $\frac{1}{9}$
C) $\f... | \frac{2}{9} | 0 | 4,670.4375 | -1 | 4,670.4375 | |
Find \( x_{1000} \) if \( x_{1} = 4 \), \( x_{2} = 6 \), and for any natural \( n \geq 3 \), \( x_{n} \) is the smallest composite number greater than \( 2 x_{n-1} - x_{n-2} \). | 2002 | 0 | 8,192 | -1 | 8,192 | |
How many different positive values of \( x \) will make this statement true: there are exactly 3 three-digit multiples of \( x \)? | 84 | 0.0625 | 8,114.625 | 6,954 | 8,192 | |
Given an infinite geometric sequence $\{a_n\}$, the product of its first $n$ terms is $T_n$, and $a_1 > 1$, $a_{2008}a_{2009} > 1$, $(a_{2008} - 1)(a_{2009} - 1) < 0$, determine the maximum positive integer $n$ for which $T_n > 1$. | 4016 | 0.0625 | 8,145.75 | 7,452 | 8,192 | |
In a regular tetrahedron ABCD with an edge length of 2, G is the centroid of triangle BCD, and M is the midpoint of line segment AG. The surface area of the circumscribed sphere of the tetrahedron M-BCD is __________. | 6\pi | 0.5 | 6,932.8125 | 6,078.25 | 7,787.375 | |
Convert $1729_{10}$ to base 6. | 12001_6 | 0.875 | 4,821.8125 | 4,340.357143 | 8,192 | |
In a circle, parallel chords of lengths 5, 12, and 13 determine central angles of $\theta$, $\phi$, and $\theta + \phi$ radians, respectively, where $\theta + \phi < \pi$. If $\sin \theta$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | 18 | 0 | 6,420.5625 | -1 | 6,420.5625 | |
What is the area enclosed by the graph of $|x| + |3y| = 9$? | 54 | 1 | 3,422.6875 | 3,422.6875 | -1 | |
The equation $x^{2}+2 x=i$ has two complex solutions. Determine the product of their real parts. | \frac{1-\sqrt{2}}{2} | The solutions are $x+1= \pm e^{\frac{i \pi}{8}} \sqrt[4]{2}$. The desired product is then $$\left(-1+\cos \left(\frac{\pi}{8}\right) \sqrt[4]{2}\right)\left(-1-\cos \left(\frac{\pi}{8}\right) \sqrt[4]{2}\right)=1-\cos ^{2}\left(\frac{\pi}{8}\right) \sqrt{2}=1-\frac{\left(1+\cos \left(\frac{\pi}{4}\right)\right)}{2} \sq... | 0 | 8,008.1875 | -1 | 8,008.1875 |
The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer. | 15 | 1 | 2,271.0625 | 2,271.0625 | -1 | |
Jenny places a total of 30 red Easter eggs in several green baskets and a total of 45 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket? | 15 | 1 | 5,577.125 | 5,577.125 | -1 | |
In a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes, each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are considered the same). The number of situa... | 18 | 0.75 | 7,124.25 | 6,768.333333 | 8,192 | |
In the rectangular coordinate system $xoy$, the parametric equations of the curve $C$ are $x=3\cos \alpha$ and $y=\sin \alpha$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of the line $l$ is $\rho \sin (\t... | \frac{18\sqrt{2}}{5} | 0 | 6,418.75 | -1 | 6,418.75 | |
Given the function $f(x)=\sin (\omega x+\varphi)$ $(0 < \omega < 3,0 < \varphi < \pi)$, if $x=-\frac{\pi}{4}$ is a zero of the function $f(x)$, and $x=\frac{\pi}{3}$ is an axis of symmetry for the graph of the function $f(x)$, then the value of $\omega$ is \_\_\_\_. | \frac{6}{7} | 0.3125 | 7,388.75 | 5,621.6 | 8,192 | |
The quantity $\tan 7.5^\circ$ can be expressed in the form
\[\tan 7.5^\circ = \sqrt{a} - \sqrt{b} + \sqrt{c} - d,\]where $a \ge b \ge c \ge d$ are positive integers. Find $a + b + c + d.$ | 13 | 0.875 | 5,880.9375 | 5,550.785714 | 8,192 | |
Jake will roll two standard six-sided dice and make a two-digit number from the numbers he rolls. If he rolls a 4 and a 2, he can form either 42 or 24. What is the probability that he will be able to make an integer between 30 and 40, inclusive? Express your answer as a common fraction. | \frac{11}{36} | 0.25 | 7,633.75 | 7,095.25 | 7,813.25 | |
Consider a 9x9 chessboard where the squares are labelled from a starting square at the bottom left (1,1) increasing incrementally across each row to the top right (9,9). Each square at position $(i,j)$ is labelled with $\frac{1}{i+j-1}$. Nine squares are chosen such that there is exactly one chosen square in each row a... | \frac{1}{362880} | 0 | 8,192 | -1 | 8,192 | |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | 40\pi | 0.375 | 6,546.3125 | 3,803.5 | 8,192 | |
In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can be repeated) so that the resulting 12-digit number is divisible by 75. How many ways can this be done? | 2592 | 0.125 | 7,765.25 | 6,014.5 | 8,015.357143 | |
Given quadrilateral ABCD, ∠A = 120∘, and ∠B and ∠D are right angles. Given AB = 13 and AD = 46, find the length of AC. | 62 | 0.875 | 4,884.625 | 4,470.928571 | 7,780.5 | |
You have four tiles marked X and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that no two X tiles are adjacent to each other? | \frac{1}{35} | 0.1875 | 7,653.875 | 5,322 | 8,192 | |
I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point? | 14 | 0.4375 | 3,810.0625 | 1,843.142857 | 5,339.888889 | |
Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square. | \frac{3+\sqrt{3}}{2} | Consider the diagram above. We see that the shape is a square plus 3 triangles. The top and bottom triangles have base $\sqrt{2}$ and height $\frac{1}{2}(\sqrt{3}-\sqrt{2})$, and the triangle on the side has the same base and height $1-\frac{\sqrt{2}}{2}$. Adding their areas, we get the answer. | 0 | 8,192 | -1 | 8,192 |
If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\sin(\pi x)$ ($0<x<2$), find the smallest positive period for $y=\tan\left(\frac{(a+b)x}{2}\right)$. | 2\pi | 0.9375 | 2,762.75 | 2,838.333333 | 1,629 | |
For the graph of a certain quadratic $y = ax^2 + bx + c$, the vertex of the parabola is $(3,7)$ and one of the $x$-intercepts is $(-2,0)$. What is the $x$-coordinate of the other $x$-intercept? | 8 | 0.9375 | 2,477.5625 | 2,096.6 | 8,192 | |
If an integer $a$ ($a \neq 1$) makes the solution of the linear equation in one variable $ax-3=a^2+2a+x$ an integer, then the sum of all integer roots of this equation is. | 16 | 0 | 4,971 | -1 | 4,971 | |
While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\%$ of his bill and Joe tipped $20\%$ of his bill. What was the difference, in dollars between their bills? | 10 | 1. **Define the variables:**
Let $m$ represent Mike's bill and $j$ represent Joe's bill.
2. **Set up the equations based on the given percentages:**
- Mike tipped $10\%$ of his bill, which is $2$ dollars. Therefore, we have the equation:
\[
\frac{10}{100}m = 2
\]
- Joe tipped $20\%$ of his bill... | 1 | 1,247.6875 | 1,247.6875 | -1 |
Given that Three people, A, B, and C, are applying to universities A, B, and C, respectively, where each person can only apply to one university, calculate the conditional probability $P\left(A|B\right)$. | \frac{1}{2} | 0.625 | 5,736.625 | 5,224 | 6,591 | |
The diagram shows a right-angled triangle \( ACD \) with a point \( B \) on the side \( AC \). The sides of triangle \( ABD \) have lengths 3, 7, and 8. What is the area of triangle \( BCD \)? | 2\sqrt{3} | 0.1875 | 7,536.0625 | 5,910.666667 | 7,911.153846 | |
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
\[\begin{aligned} ax+by&=1 \\ x^2 + y^2 &= 50 \end{aligned}\]has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there? | 72 | 0 | 8,174.125 | -1 | 8,174.125 | |
Let \(a, b, c, d\) be positive integers such that \(a^5 =\) | 757 | 0 | 7,941.9375 | -1 | 7,941.9375 | |
Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/2000$. Find $n$. | 26 | 0.5 | 7,180.3125 | 6,168.625 | 8,192 | |
What is the degree measure of angle $LOQ$ when polygon $\allowbreak LMNOPQ$ is a regular hexagon? [asy]
draw((-2,0)--(-1,1.73205081)--(1,1.73205081)--(2,0)--(1,-1.73205081)--(-1,-1.73205081)--cycle);
draw((-1,-1.73205081)--(1,1.73205081)--(1,-1.73205081)--cycle);
label("L",(-1,-1.73205081),SW);
label("M",(-2,0),W);
lab... | 30^\circ | 0.5625 | 7,011.875 | 6,202.111111 | 8,053 | |
In triangle $DEF$, we have $\angle D = 90^\circ$ and $\sin E = \frac{3}{5}$. Find $\cos F$. | \frac{3}{5} | 0.9375 | 3,844.0625 | 3,554.2 | 8,192 | |
How many four-digit integers are divisible by both 7 and 5? | 257 | 0.9375 | 3,526.625 | 3,215.6 | 8,192 | |
Through the vertices \( A, C, D \) of the parallelogram \( ABCD \) with sides \( AB = 7 \) and \( AD = 4 \), a circle is drawn that intersects the line \( BD \) at point \( E \), and \( DE = 13 \). Find the length of diagonal \( BD \). | 15 | 0 | 8,118.0625 | -1 | 8,118.0625 | |
Given that $α$ is an angle in the third quadrant and $\cos 2α=-\frac{3}{5}$, find $\tan (\frac{π}{4}+2α)$. | -\frac{1}{7} | 0.5625 | 6,027.875 | 5,065.666667 | 7,265 | |
In what ratio does the angle bisector of the acute angle divide the area of a right trapezoid inscribed in a circle? | 1:1 | 0.125 | 7,961 | 6,344 | 8,192 | |
A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form
\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$ | 49x^2 - 42xy + 9y^2 - 222x - 54y + 603 = 0 | 0.3125 | 5,645.0625 | 3,235 | 6,740.545455 | |
Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor {\sqrt{x}} \rfloor = 12$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 120$. ($\lfloor {v} \rfloor$ means the greatest integer less than or equal to $v$.) | \frac{241}{2500} | 1. **Determine the range for $x$ based on $\lfloor \sqrt{x} \rfloor = 12$:**
Since $\lfloor \sqrt{x} \rfloor = 12$, it implies that $12 \leq \sqrt{x} < 13$. Squaring both sides of the inequality, we get:
\[
12^2 \leq x < 13^2 \implies 144 \leq x < 169
\]
2. **Determine the range for $x$ based on $\lfloor \... | 0.9375 | 3,735.8125 | 3,713.733333 | 4,067 |
There are 54 students in a class, and there are 4 tickets for the Shanghai World Expo. Now, according to the students' ID numbers, the tickets are distributed to 4 students through systematic sampling. If it is known that students with ID numbers 3, 29, and 42 have been selected, then the ID number of another student w... | 16 | 0.8125 | 4,485.125 | 3,629.692308 | 8,192 | |
Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously? | 35 | 0 | 8,093.9375 | -1 | 8,093.9375 | |
There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.\] Find $n_1 + n_2 + \cdots + n_r$. | 21 | In base $3$, we find that $\overline{2008}_{10} = \overline{2202101}_{3}$. In other words,
$2008 = 2 \cdot 3^{6} + 2 \cdot 3^{5} + 2 \cdot 3^3 + 1 \cdot 3^2 + 1 \cdot 3^0$
In order to rewrite as a sum of perfect powers of $3$, we can use the fact that $2 \cdot 3^k = 3^{k+1} - 3^k$:
$2008 = (3^7 - 3^6) + (3^6-3^5) + (3... | 0.25 | 7,990.25 | 7,385 | 8,192 |
The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum
\[ \frac{1}{1-p} + \frac{1}{1-q} +\frac{1}{1-r} ?\] | 12 | 0.6875 | 5,197.125 | 3,835.818182 | 8,192 | |
Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms. | 44 | Using Simon's Favorite Factoring Trick, we rewrite as $[(x+7)(y-3)]^n = (x+7)^n(y-3)^n$. Both binomial expansions will contain $n+1$ non-like terms; their product will contain $(n+1)^2$ terms, as each term will have an unique power of $x$ or $y$ and so none of the terms will need to be collected. Hence $(n+1)^2 \ge 199... | 0.8125 | 4,589.5 | 3,758.153846 | 8,192 |
Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? | 64 | We are given that each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. We need to find the total number of students and rabbits in all $4$ classrooms and then determine how many more students there are than rabbits.
#### Step 1: Calculate the total number of students in all class... | 1 | 971.625 | 971.625 | -1 |
Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and 1 form the side lengths of an obtuse triangle. | \frac{\pi-2}{4} | We require $a+b>1$ and $a^{2}+b^{2}<1$. Geometrically, this is the area enclosed in the quarter-circle centered at the origin with radius 1, not including the area enclosed by $a+b<1$ (an isosceles right triangle with side length 1). As a result, our desired probability is $\frac{\pi-2}{4}$. | 0.25 | 7,075.875 | 6,022.25 | 7,427.083333 |
Let $ABC$ be a triangle with $\angle BAC=40^\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\parallel OG$ , find $\angle ACB$ . | 70 | 0 | 8,192 | -1 | 8,192 | |
Twenty switches in an office computer network are to be connected so that each switch has a direct connection to exactly three other switches. How many connections will be necessary? | 30 | 1 | 1,587.1875 | 1,587.1875 | -1 | |
Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\triangle DEF$ to the area of quadrilateral $ABEF$ is: | 1: 5 | 0 | 6,083.4375 | -1 | 6,083.4375 | |
Simplify $2(3-i)+i(2+i)$. | 5 | 1 | 1,894.8125 | 1,894.8125 | -1 | |
Find the next two smallest juicy numbers after 6, and show a decomposition of 1 into unit fractions for each of these numbers. | 12, 15 | 12 and 15: $1=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{12}$, $1=\frac{1}{2}+\frac{1}{3}+\frac{1}{10}+\frac{1}{15}$. | 0 | 7,583.75 | -1 | 7,583.75 |
Given that $P$ is a moving point on the curve $y= \frac {1}{4}x^{2}- \frac {1}{2}\ln x$, and $Q$ is a moving point on the line $y= \frac {3}{4}x-1$, then the minimum value of $PQ$ is \_\_\_\_\_\_. | \frac {2-2\ln 2}{5} | 0 | 7,912.125 | -1 | 7,912.125 | |
In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks must be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can be repeated) so that the resulting 10-digit number is divisible by 18. How many ways can this be done? | 3645 | 0.3125 | 7,200.1875 | 5,592.6 | 7,930.909091 | |
Let $P$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_6x_1$ where $x_1, x_2, \dots, x_6$ is a permutation of $(1,2,3,4,5,6)$ and let $Q$ be the number of permutations for which this maximum is achieved, given the additional condition that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 21$. Evaluate $P + Q$. | 83 | 0.0625 | 8,171.875 | 7,870 | 8,192 | |
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | 32 | 0.3125 | 7,529.75 | 6,439 | 8,025.545455 | |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as t... | \frac{86}{7} | 0.9375 | 5,750.4375 | 5,587.666667 | 8,192 | |
Find the number of integers $n$ with $1 \leq n \leq 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of 1001. | 99 | Note that $1001=7 \cdot 11 \cdot 13$, so the stated product must be a multiple of 7, as well as a multiple of 11, as well as a multiple of 13. There are 4 possible residues of $n$ modulo 11 for which the product is a multiple of 11; similarly, there are 4 possible residues of $n$ modulo 13 for which the product is a mu... | 0.0625 | 7,983.125 | 8,192 | 7,969.2 |
The equation $z^6+z^3+1=0$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$. | 160 | The substitution $y=z^3$ simplifies the equation to $y^2+y+1 = 0$. Applying the quadratic formula gives roots $y=-\frac{1}{2}\pm \frac{\sqrt{3}i}{2}$, which have arguments of $120$ and $240,$ respectively. We can write them as $z^3 = \cos 240^\circ + i\sin 240^\circ$ and $z^3 = \cos 120^\circ + i\sin 120^\circ$. So we ... | 0.9375 | 5,080.625 | 4,873.2 | 8,192 |
Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.
One day, Uncle Chernomor asked each warrior four questions:
- Is your favorite weapon a sword?
- Is your favorite weapon a spear?
- Is your favorite w... | 12 | 0.125 | 7,700.75 | 4,262 | 8,192 | |
Inside a square with side length 12, two congruent equilateral triangles are drawn such that each has one vertex touching two adjacent vertices of the square and they share one side. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles? | 12 - 4\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
We have an equilateral triangle with circumradius $1$ . We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$ , is maximum.
(The total distance of the point P from the sides of an equilatera... | 3\sqrt{3} | 0.125 | 7,863.5 | 6,928.5 | 7,997.071429 | |
The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$ , where $n$ and $k$ are integers and $0\leq k<2013$ . What is $k$ ? Recall $2013=3\cdot 11\cdot 61$ . | 27 | 0.8125 | 5,289.6875 | 4,811.846154 | 7,360.333333 | |
In a class at school, all students are the same age, except seven of them who are 1 year younger and two of them who are 2 years older. The sum of the ages of all the students in this class is 330. How many students are in this class? | 37 | 0.5625 | 7,438.125 | 6,851.777778 | 8,192 | |
Consider a 6x3 grid where you can move only to the right or down. How many valid paths are there from top-left corner $A$ to bottom-right corner $B$, if paths passing through segment from $(4,3)$ to $(4,2)$ and from $(2,1)$ to $(2,0)$ are forbidden? [The coordinates are given in usual (x, y) notation, where the leftmos... | 48 | 0 | 6,392.5625 | -1 | 6,392.5625 | |
A rectangular table of dimensions \( x \) cm \(\times 80\) cm is covered with identical sheets of paper of size \( 5 \) cm \(\times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet ... | 77 | 0.25 | 7,158.125 | 4,945 | 7,895.833333 | |
How many natural numbers greater than 6 but less than 60 are relatively prime to 15? | 29 | 0.9375 | 5,441.5 | 5,258.133333 | 8,192 | |
The diagram shows a square, its two diagonals, and two line segments, each of which connects two midpoints of the sides of the square. What fraction of the area of the square is shaded?
A) $\frac{1}{8}$
B) $\frac{1}{10}$
C) $\frac{1}{12}$
D) $\frac{1}{16}$
E) $\frac{1}{24}$ | \frac{1}{16} | 0 | 8,032.75 | -1 | 8,032.75 | |
Given the square of an integer $x$ is 1521, what is the value of $(x+1)(x-1)$? | 1520 | 1 | 1,028 | 1,028 | -1 | |
Given the numbers \( x, y, z \in [0, \pi] \), find the minimum value of the expression
$$
A = \cos (x - y) + \cos (y - z) + \cos (z - x)
$$ | -1 | 0.375 | 8,007.875 | 7,701 | 8,192 | |
In triangle $\triangle ABC$, $sin(A+\frac{π}{4})sin(B+\frac{π}{4})=cosAcosB$. Find:<br/>
$(1)$ the value of angle $C$;<br/>
$(2)$ if $AB=\sqrt{2}$, find the minimum value of $\overrightarrow{CA}•\overrightarrow{CB}$. | -\sqrt{2}+1 | 0 | 6,941.3125 | -1 | 6,941.3125 | |
The attached figure is an undirected graph. The circled numbers represent the nodes, and the numbers along the edges are their lengths (symmetrical in both directions). An Alibaba Hema Xiansheng carrier starts at point A and will pick up three orders from merchants B_{1}, B_{2}, B_{3} and deliver them to three customer... | 16 | The shortest travel distance is 16, attained by the carrier taking the following stops: A \rightsquigarrow B_{2} \rightsquigarrow C_{2} \rightsquigarrow B_{1} \rightsquigarrow B_{3} \rightsquigarrow C_{3} \rightsquigarrow C_{1}. There are two slightly different routes with the same length of 16: Route 1: 2(A) \rightarr... | 0 | 7,694.0625 | -1 | 7,694.0625 |
Let \[f(x) = \left\{
\begin{array}{cl}
\frac{x}{21} & \text{ if }x\text{ is a multiple of 3 and 7}, \\
3x & \text{ if }x\text{ is only a multiple of 7}, \\
7x & \text{ if }x\text{ is only a multiple of 3}, \\
x+3 & \text{ if }x\text{ is not a multiple of 3 or 7}.
\end{array}
\right.\]If $f^a(x)$ means the function is n... | 7 | 0.375 | 3,262.375 | 3,177.666667 | 3,313.2 | |
In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. It is given that \\(a+b=5\\), \\(c=\sqrt{7}\\), and \\(4{{\left( \sin \frac{A+B}{2} \right)}^{2}}-\cos 2C=\frac{7}{2}\\).
\\((1)\\) Find the magnitude of angle \\(C\\);
\\((2)\\) ... | \frac {3 \sqrt {3}}{2} | 0 | 3,251.6875 | -1 | 3,251.6875 | |
In a board game, I move on a linear track. For move 1, I stay still. For subsequent moves $n$ where $2 \le n \le 30$, I move forward two steps if $n$ is prime and three steps backward if $n$ is composite. How many steps in total will I need to make to return to my original starting position after all 30 moves? | 37 | 0.25 | 6,462.3125 | 6,394.25 | 6,485 | |
The equations of the asymptotes of the hyperbola $\frac{x^2}{2}-y^2=1$ are ________, and its eccentricity is ________. | \frac{\sqrt{6}}{2} | 0 | 1,671.125 | -1 | 1,671.125 | |
Determine the value of $x$ for which $10^x \cdot 500^{x} = 1000000^{3}$.
A) $\frac{9}{1.699}$
B) $6$
C) $\frac{18}{3.699}$
D) $5$
E) $20$ | \frac{18}{3.699} | 0 | 6,642.0625 | -1 | 6,642.0625 | |
Fran writes the numbers \(1,2,3, \ldots, 20\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \(n\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \(n\) that are still on the chalkboard (inc... | \frac{131}{10} | For each \(n, 1 \leq n \leq 20\), consider the first time that Fran chooses one of the multiples of \(n\). It is in this move that \(n\) is erased, and all the multiples of \(n\) at most 20 are equally likely to be chosen for this move. Hence this is the only move in which Fran could possibly choose \(n\); since there ... | 0 | 8,192 | -1 | 8,192 |
Find the number of subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that are subsets of neither $\{1, 2, 3, 4, 5\}$ nor $\{4, 5, 6, 7, 8\}$. | 196 | Note that by Principle of Inclusion and Exclusion, the total number of subsets must be $2^8-2^5-2^5+2^2$ as denoted by above. Thus our answer is $64(3)+4 = \boxed{196}$ | 0.875 | 5,925.5625 | 5,601.785714 | 8,192 |
What is the value of $\frac13\cdot\frac92\cdot\frac1{27}\cdot\frac{54}{1}\cdot\frac{1}{81}\cdot\frac{162}{1}\cdot\frac{1}{243}\cdot\frac{486}{1}$? | 12 | 0.4375 | 6,840.4375 | 5,713.428571 | 7,717 | |
Let $x_1$ , $x_2$ , …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$ ? | 171 | 0.75 | 5,932.3125 | 5,179.083333 | 8,192 | |
A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? | 256 | 0.9375 | 997 | 1,045.066667 | 276 | |
Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
Find the value of \( f(90) \). | 999 | 0 | 8,192 | -1 | 8,192 | |
Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of this hyperbola, find the x-coordinate of \( P \). | -\frac{64}{5} | 0.0625 | 8,090.1875 | 6,563 | 8,192 | |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 | 0.25 | 6,462.8125 | 5,848.25 | 6,667.666667 | |
A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed? | 18\% | 0.8125 | 4,025.625 | 3,064.153846 | 8,192 | |
In the diagram, the number line between 0 and 2 is divided into 8 equal parts. The numbers 1 and \(S\) are marked on the line. What is the value of \(S\)? | 1.25 | 0.25 | 663.625 | 462.25 | 730.75 | |
A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. | 64 | 0.1875 | 7,367.6875 | 5,407 | 7,820.153846 | |
Define the operation "" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \lg|x + 2|$ (where $x \neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$. | -8 | 0.75 | 6,659.75 | 6,149 | 8,192 | |
Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points? | 3 | Note that both points must be in different semicircles to reach the maximum distance. Let these points be $M$ and $N$, and $O_{1}$ and $O_{2}$ be the centers of the two semicircles where they lie respectively. Then $$M N \leq M O_{1}+O_{1} O_{2}+O_{2} N$$ Note that the the right side will always be equal to 3 ($M O_{1}... | 0 | 8,192 | -1 | 8,192 |
Find, with proof, the number of positive integers whose base- $n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.) | \[ 2^{n+1} - 2(n+1) \] | Let a $k$ -good sequence be a sequence of distinct integers $\{ a_i \}_{i=1}^k$ such that for all integers $2\le i \le k$ , $a_i$ differs from some preceding term by $\pm 1$ .
Lemma. Let $a$ be an integer. Then there are $2^{k-1}$ $k$ -good sequences starting on $a$ , and furthermore, the terms of each of these seque... | 0 | 8,192 | -1 | 8,192 |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Find the sum of the slope and $y$-intercept of the line passing through the points $C$ and $D.$ [asy]
size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,6);
B=(0,0);
C=(8,0);
D=(0,3);
E=(4,... | \frac{21}{8} | 1 | 1,852.25 | 1,852.25 | -1 | |
For any positive integers $a$ and $b$ with $b>1$, let $s_{b}(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\sum_{i=1}^{\left\lfloor\log _{23} n\right\rfloor} s_{20}\left(\left\lfloor\frac{n}{23^{i}}\right\rfloor\right)=103 \quad \text { and } \sum_{i=1... | 81 | First we will prove that $$s_{a}(n)=n-(a-1)\left(\sum_{i=1}^{\infty}\left\lfloor\frac{n}{a^{i}}\right\rfloor\right)$$ If $n=\left(n_{k} n_{k-1} \cdots n_{1} n_{0}\right)_{a}$, then the digit $n_{i}$ contributes $n_{i}$ to the left side of the sum, while it contributes $$n_{i}\left(a^{i}-(a-1)\left(a^{i-1}+a^{i-2}+\cdot... | 0 | 8,192 | -1 | 8,192 |
Regular octagonal pyramid $\allowbreak PABCDEFGH$ has the octagon $ABCDEFGH$ as its base. Each side of the octagon has length 5. Pyramid $PABCDEFGH$ has an additional feature where triangle $PAD$ is an equilateral triangle with side length 10. Calculate the volume of the pyramid. | \frac{250\sqrt{3}(1 + \sqrt{2})}{3} | 0 | 7,908.125 | -1 | 7,908.125 | |
On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees? | 9 | 1 | 2,156.8125 | 2,156.8125 | -1 |
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