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 |
|---|---|---|---|---|---|---|
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is *monotonically bounded* if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$ . We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a *mountain* if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$ . Ev... | 49153 | 0 | 8,192 | -1 | 8,192 | |
A store has equal amounts of candies priced at 2 rubles per kilogram and candies priced at 3 rubles per kilogram. At what price should the mixture of these candies be sold? | 2.4 | 0 | 1,372.6875 | -1 | 1,372.6875 | |
In a $7 \times 7$ grid, choose $k$ cells such that the centers of any 4 chosen cells do not form the vertices of a rectangle. Find the maximum value of $k$ that satisfies this condition. | 21 | 0.0625 | 7,826.6875 | 4,452 | 8,051.666667 | |
Jillian drives along a straight road that goes directly from her house $(J)$ to her Grandfather's house $(G)$. Some of this road is on flat ground and some is downhill or uphill. Her car travels downhill at $99 \mathrm{~km} / \mathrm{h}$, on flat ground at $77 \mathrm{~km} / \mathrm{h}$, and uphill at $63 \mathrm{~km} ... | 308 | 0.5 | 7,150.0625 | 6,108.125 | 8,192 | |
Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel? | 2\sqrt{41} | 1 | 2,215.3125 | 2,215.3125 | -1 | |
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. | 619 | 0.25 | 6,898.25 | 4,805.5 | 7,595.833333 | |
A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run at least to catch up with the rabbit? | 240 | 0.0625 | 8,099.4375 | 6,711 | 8,192 | |
In how many ways can the number 1024 be factored into three natural factors such that the first factor is a multiple of the second, and the second is a multiple of the third? | 14 | 0.5625 | 7,171.1875 | 6,505.666667 | 8,026.857143 | |
Given that the probability mass function of the random variable $X$ is $P(X=k)= \frac{k}{25}$ for $k=1, 2, 3, 4, 5$, find the value of $P(\frac{1}{2} < X < \frac{5}{2})$. | \frac{1}{5} | 0 | 2,903.125 | -1 | 2,903.125 | |
In the diagram, $AB$ is parallel to $DC,$ and $ACE$ is a straight line. What is the value of $x?$ [asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)--cycle);
draw((-.5,5)--(8.5,-10/7));
label("$A$",(-.5,5),W);
label("$B$",(8,5),E);
label("$C$",(6.5,0),S);
label("$D$",(0,0),SW);
label("$E$",(8.5,-10/7),S);
draw((2,0)--(3,0),Arro... | 35 | 0 | 6,556.875 | -1 | 6,556.875 | |
Points with integer coordinates (including zero) are called lattice points (or grid points). Find the total number of lattice points (including those on the boundary) in the region bounded by the x-axis, the line \(x=4\), and the parabola \(y=x^2\). | 35 | 0.8125 | 4,956.1875 | 4,209.461538 | 8,192 | |
In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is... | 0.75 | 0 | 8,122.625 | -1 | 8,122.625 | |
The sum of seven test scores has a mean of 84, a median of 85, and a mode of 88. Calculate the sum of the three highest test scores. | 264 | 0.625 | 7,274.9375 | 6,724.7 | 8,192 | |
Given that $x$ and $y$ are positive integers, and $x^2 - y^2 = 53$, find the value of $x^3 - y^3 - 2(x + y) + 10$. | 2011 | 0.125 | 5,659.5 | 4,264 | 5,858.857143 | |
Given $a+b+c=0$ and $a^2+b^2+c^2=1$, find the values of $ab+bc+ca$ and $a^4+b^4+c^4$. | \frac{1}{2} | 0.0625 | 7,904.5 | 3,592 | 8,192 | |
The sequence $ (a_n)$ is given by $ a_1\equal{}1,a_2\equal{}0$ and:
$ a_{2k\plus{}1}\equal{}a_k\plus{}a_{k\plus{}1}, a_{2k\plus{}2}\equal{}2a_{k\plus{}1}$ for $ k \in \mathbb{N}.$
Find $ a_m$ for $ m\equal{}2^{19}\plus{}91.$ | 91 | 0 | 8,192 | -1 | 8,192 | |
How many three-digit numbers remain if we exclude all three-digit numbers in which all digits are the same or the middle digit is different from the two identical end digits? | 810 | 0.375 | 6,620.3125 | 5,640.5 | 7,208.2 | |
On a plate, there are different candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Sveta ate all of them one by one, choosing each next candy at random. Find the probability that the first and last candies she ate were of the same type. | 14/45 | 0.4375 | 6,769.75 | 4,941.142857 | 8,192 | |
An isosceles triangle $ABP$ with sides $AB = AP = 3$ inches and $BP = 4$ inches is placed inside a square $AXYZ$ with a side length of $8$ inches, such that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. Cal... | \frac{32\pi}{3} | 0 | 8,108.625 | -1 | 8,108.625 | |
The midsegment of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. Given that the height of the trapezoid is 2, what is the area of this trapezoid? | 148 | 0.5 | 6,289 | 4,386 | 8,192 | |
The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units, and the resulting graph corresponds to an even function. Find the minimum value of $m$ such that there exists $x \in \left[ 0,\frac{\pi}{2} \right]$ such that the inequality $f(x) \leqslant... | -\frac{1}{2} | 0.5625 | 6,715.0625 | 6,286.555556 | 7,266 | |
Given that two children, A and B, and three adults, 甲, 乙, and 丙, are standing in a line, A is not at either end, and exactly two of the three adults are standing next to each other. The number of different arrangements is $\boxed{\text{answer}}$. | 48 | 0 | 8,192 | -1 | 8,192 | |
Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\] | \frac{\sqrt{373}}{6} | 0 | 6,279.625 | -1 | 6,279.625 | |
How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable and neither are the boxes? | 5 | 0.6875 | 4,996.5625 | 3,996.727273 | 7,196.2 | |
If $x = 2y$ and $y \neq 0$, what is the value of $(x-y)(2x+y)$? | 5y^{2} | Since $x = 2y$, then $(x-y)(2x+y) = (2y-y)(2(2y)+y) = (y)(5y) = 5y^{2}$. | 0 | 1,535.3125 | -1 | 1,535.3125 |
Find the smallest positive multiple of 9 that can be written using only the digits: (a) 0 and 1; (b) 1 and 2. | 12222 | 0.3125 | 7,274.75 | 6,294.6 | 7,720.272727 | |
For how many real values of $c$ do we have $\left|\frac12-ci\right| = \frac34$? | 2 | 1 | 1,771.125 | 1,771.125 | -1 | |
The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18. | 12 | 0.5 | 6,348.0625 | 4,504.125 | 8,192 | |
From the set $\{1, 2, 3, \ldots, 10\}$, select 3 different elements such that the sum of these three numbers is a multiple of 3, and the three numbers cannot form an arithmetic sequence. Calculate the number of ways to do this. | 22 | 0.625 | 6,580.0625 | 5,962.4 | 7,609.5 | |
Three positive integers $a,$ $b,$ and $x$ form an O'Hara triple $(a,b,x)$ if $\sqrt{a}+\sqrt{b}=x.$ For example, $(1,4,3)$ is an O'Hara triple because $\sqrt{1}+\sqrt{4}=3.$
If $(36,25,x)$ is an O'Hara triple, determine the value of $x.$ | 11 | 1 | 376.875 | 376.875 | -1 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $4\times 4$ square array of dots, as in the figure below?
[asy]size(2cm,2cm); for (int i=0; i<4; ++i) { for (int j=0; j<4; ++j) { filldraw(Circle((i, j), .05), black, black); } } [/asy] (Two rectangles are di... | 36 | 0.125 | 7,649.375 | 6,037 | 7,879.714286 | |
Compute $\sum_{k=1}^{2009} k\left(\left\lfloor\frac{2009}{k}\right\rfloor-\left\lfloor\frac{2008}{k}\right\rfloor\right)$. | 2394 | The summand is equal to $k$ if $k$ divides 2009 and 0 otherwise. Thus the sum is equal to the sum of the divisors of 2009, or 2394. | 0.4375 | 6,134.25 | 4,078.857143 | 7,732.888889 |
Given vectors $\overrightarrow{m}=(\sqrt{3}\cos x,-\cos x)$ and $\overrightarrow{n}=(\cos (x-\frac{π}{2}),\cos x)$, satisfying the function $f\left(x\right)=\overrightarrow{m}\cdot \overrightarrow{n}+\frac{1}{2}$.
$(1)$ Find the interval on which $f\left(x\right)$ is monotonically increasing on $[0,\frac{π}{2}]$.
$... | \frac{12\sqrt{3}-5}{26} | 0 | 5,752.875 | -1 | 5,752.875 | |
A bus arrives randomly sometime between 1:00 and 2:30, waits for 20 minutes, and then leaves. If Laura also arrives randomly between 1:00 and 2:30, what is the probability that the bus will be there when Laura arrives? | \frac{16}{81} | 0.375 | 6,984.3125 | 5,269 | 8,013.5 | |
Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degre... | 120^\circ | 0.75 | 6,185.1875 | 5,516.25 | 8,192 | |
Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them forgot gloves at the zoo, 52 forgot scarves, and 50 managed to forget hats. Find the smallest number of the most scatterbrained students - those who lost all three items. | 37 | 0.625 | 6,697.375 | 6,031.9 | 7,806.5 | |
Given that $\cos \alpha =\dfrac{\sqrt{5}}{5}$ and $\sin (\alpha -\beta )=\dfrac{\sqrt{10}}{10}$, calculate the value of $\cos \beta$. | \dfrac{\sqrt{2}}{2} | 0 | 7,414.125 | -1 | 7,414.125 | |
Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.
If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt3/... | 803 | When $n>1$, the path of blocks has $6(n-1)$ blocks total in it. When $n=1$, there is just one lonely block. Thus, the area of the garden enclosed by the path when $n=202$ is
\[(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A\],
where $A$ is the area of one block. Then, because $n(n+1)/2$ is equal to the sum of the first... | 0 | 4,333.9375 | -1 | 4,333.9375 |
Let $D$ be the circle with the equation $2x^2 - 8y - 6 = -2y^2 - 8x$. Determine the center $(c,d)$ of $D$ and its radius $s$, and calculate the sum $c + d + s$. | \sqrt{7} | 0 | 3,816.8125 | -1 | 3,816.8125 | |
Let $p$, $q$, $r$, $s$, and $t$ be positive integers with $p+q+r+s+t=2025$ and let $N$ be the largest of the sums $p+q$, $q+r$, $r+s$, and $s+t$. Determine the smallest possible value of $N$. | 676 | 0.0625 | 7,526.5625 | 6,808 | 7,574.466667 | |
Let \( S = \{1, 2, \cdots, 2009\} \). \( A \) is a 3-element subset of \( S \) such that all elements in \( A \) form an arithmetic sequence. How many such 3-element subsets \( A \) are there? | 1008016 | 0.5 | 6,706.625 | 5,221.25 | 8,192 | |
Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 3,$ $\|\mathbf{w}\| = 7,$ and $\mathbf{v} \cdot \mathbf{w} = 10,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.$ | \frac{10}{7} | 1 | 1,524.875 | 1,524.875 | -1 | |
If $\sum_{n = 0}^{\infty}\sin^{2n}\theta = 4$, what is the value of $\sin{2\theta}$? | \frac{\sqrt{3}}{2} | 0 | 7,779.25 | -1 | 7,779.25 | |
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meter... | 17 | 1. **Establishing the Coordinate System and Points**:
Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be calculated based on the geometry of a regular hexagon:
... | 0.8125 | 4,921.3125 | 4,581.076923 | 6,395.666667 |
Given a function $f(x)$ satisfies $f(x) + f(4-x) = 4$, $f(x+2) - f(-x) = 0$, and $f(1) = a$, calculate the value of $f(1) + f(2) + f(3) + \cdots + f(51)$. | 102 | 0.875 | 5,920.875 | 5,596.428571 | 8,192 | |
If Xiao Zhang's daily sleep time is uniformly distributed between 6 to 9 hours, what is the probability that his average sleep time over two consecutive days is at least 7 hours? | 7/9 | 0.1875 | 7,643.4375 | 6,220.666667 | 7,971.769231 | |
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positi... | 26 | 0.9375 | 5,620.625 | 5,449.2 | 8,192 | |
Joe has a rectangular lawn measuring 120 feet by 180 feet. His lawn mower has a cutting swath of 30 inches, and he overlaps each cut by 6 inches to ensure no grass is missed. Joe mows at a rate of 4000 feet per hour. Calculate the time it will take Joe to mow his entire lawn. | 2.7 | 0.0625 | 1,960.8125 | 859 | 2,034.266667 | |
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order (each digit is greater than the previous digit)? | 34 | 0.8125 | 5,264.125 | 4,588.461538 | 8,192 | |
Six teams play in a soccer tournament where each team faces every other team exactly once. Each match results in a win or a loss, with no ties allowed. Winners receive one point; losers receive none. In the first match, Team $A$ defeats Team $B$. Assuming each team has an equal chance to win each match, and all match o... | 419 | 0.375 | 6,820.6875 | 5,552.666667 | 7,581.5 | |
Let $f(r)=\sum_{j=2}^{2008} \frac{1}{j^{r}}=\frac{1}{2^{r}}+\frac{1}{3^{r}}+\cdots+\frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$. | \frac{2007}{2008} | We change the order of summation: $$\sum_{k=2}^{\infty} \sum_{j=2}^{2008} \frac{1}{j^{k}}=\sum_{j=2}^{2008} \sum_{k=2}^{\infty} \frac{1}{j^{k}}=\sum_{j=2}^{2008} \frac{1}{j^{2}\left(1-\frac{1}{j}\right)}=\sum_{j=2}^{2008} \frac{1}{j(j-1)}=\sum_{j=2}^{2008}\left(\frac{1}{j-1}-\frac{1}{j}\right)=1-\frac{1}{2008}=\frac{20... | 1 | 2,595.625 | 2,595.625 | -1 |
A point A is a fixed point on the circumference of a circle with a perimeter of 3. If a point B is randomly selected on the circumference, the probability that the length of the minor arc is less than 1 is ______. | \frac{2}{3} | 0.4375 | 6,621.4375 | 5,296.571429 | 7,651.888889 | |
There are $5$ people arranged in a row. Among them, persons A and B must be adjacent, and neither of them can be adjacent to person D. How many different arrangements are there? | 36 | 0 | 8,192 | -1 | 8,192 | |
An engineer invested $\$10,\!000$ in a six-month savings certificate that paid a simple annual interest rate of $12\%$. After six months, she invested the total value of her investment in another six-month certificate. After six more months, the investment was worth $\$11,\!130$. If the annual interest rate of the seco... | 10 | 1 | 2,145 | 2,145 | -1 | |
Two individuals undertake a certain task and work for an equal amount of time. $A$ misses 2 days and earns 80 forints in total, while $B$ misses 5 days and earns 63 forints. If $A$ had missed 5 days and $B$ had missed 2 days, then $A$ would earn 2 forints more than $B$. How many days did the work last? | 32 | 0.5 | 6,295.0625 | 4,398.125 | 8,192 | |
If
\[\tan x = \frac{2ab}{a^2 - b^2},\]where $a > b > 0$ and $0^\circ < x < 90^\circ,$ then find $\sin x$ in terms of $a$ and $b.$ | \frac{2ab}{a^2 + b^2} | 1 | 2,356.25 | 2,356.25 | -1 | |
Given the function $f(x)=a^{x}-(k+1)a^{-x}$ where $a > 0$ and $a\neq 1$, which is an odd function defined on $\mathbb{R}$.
1. Find the value of $k$.
2. If $f(1)= \frac {3}{2}$, and the minimum value of $g(x)=a^{2x}+a^{-2x}-2mf(x)$ on $[0,+\infty)$ is $-6$, find the value of $m$. | 2 \sqrt {2} | 0 | 6,269.5625 | -1 | 6,269.5625 | |
Winnie wrote all the integers from 1 to 2017 inclusive on a board. She then erased all the integers that are a multiple of 3. Next, she reinstated all those integers that are a multiple of 6. Finally, she erased all integers then on the board which are a multiple of 27. Of the 2017 integers that began in the list, how ... | 373 | 0 | 7,110.125 | -1 | 7,110.125 | |
Three positive integers $a$, $b$, and $c$ satisfy $a\cdot b\cdot c=8!$ and $a<b<c$. What is the smallest possible value of $c-a$? | 4 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Let \(ABCD\) be a quadrilateral inscribed in a unit circle with center \(O\). Suppose that \(\angle AOB = \angle COD = 135^\circ\), and \(BC = 1\). Let \(B'\) and \(C'\) be the reflections of \(A\) across \(BO\) and \(CO\) respectively. Let \(H_1\) and \(H_2\) be the orthocenters of \(AB'C'\) and \(BCD\), respectively.... | \frac{1}{4}(8-\sqrt{6}-3\sqrt{2}) | 0 | 8,192 | -1 | 8,192 | |
Let $n$ be an integer with $n \geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$? | 2n-2 | The answer is $2n-2$. Write $p(x) = a_nx^n+\cdots+a_1x+a_0$ and $p(x)^2 = b_{2n}x^{2n}+\cdots+b_1x+b_0$. Note that $b_0 = a_0^2$ and $b_{2n} = a_n^2$. We claim that not all of the remaining $2n-1$ coefficients $b_1,\ldots,b_{2n-1}$ can be negative, whence the largest possible number of negative coefficients is $\leq 2n... | 0 | 8,192 | -1 | 8,192 |
The numbers \(a, b,\) and \(c\) (not necessarily integers) satisfy the conditions
\[
a + b + c = 0 \quad \text{and} \quad \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 100
\]
What is the value of \(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\)? | -101 | 0 | 8,064.9375 | -1 | 8,064.9375 | |
Find the remainder when $109876543210$ is divided by $180$. | 10 | 0.8125 | 5,350.1875 | 4,694.384615 | 8,192 | |
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 4 numbers? | 60 | Two opposite vertices are relatively prime, but they both share a factor with their common neighbor. So that common neighbor must have two prime factors. So each of the 4 numbers has two prime factors, which are not shared with the opposite vertex. Moreover, it suffices to choose the vertices to be the numbers ab, bc, ... | 0 | 8,151.625 | -1 | 8,151.625 |
In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is | 5/6 | 1. **Identify the shapes and their properties:**
- $A$, $E$, and $F$ are isosceles right triangles.
- $B$, $C$, and $D$ are squares with side length $1$.
- $G$ is an equilateral triangle.
2. **Analyze the geometric arrangement:**
- The squares suggest a structure based on a unit cube, as each square can be... | 0 | 8,192 | -1 | 8,192 |
Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$. Compute $ \frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$. | \frac{49}{6} | 0.8125 | 4,275.0625 | 3,771.076923 | 6,459 | |
Let \(x,\) \(y,\) \(z\) be real numbers such that \(9x^2 + 4y^2 + 25z^2 = 1.\) Find the maximum value of
\[8x + 3y + 10z.\] | \sqrt{173} | 0 | 6,027.75 | -1 | 6,027.75 | |
Given a circle $O$ with radius $1$, $PA$ and $PB$ are two tangents to the circle, and $A$ and $B$ are the points of tangency. The minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is \_\_\_\_\_\_. | -3+2\sqrt{2} | 0 | 7,636 | -1 | 7,636 | |
Given the function $f(x)=\frac{ax+b}{{x}^{2}+4}$ attains a maximum value of $1$ at $x=-1$, find the minimum value of $f(x)$. | -\frac{1}{4} | 0.875 | 4,162.75 | 3,748.928571 | 7,059.5 | |
How many different graphs with 9 vertices exist where each vertex is connected to 2 others? | 4 | It suffices to consider the complements of the graphs, so we are looking for graphs with 9 vertices, where each vertex is connected to 2 others. There are $\mathbf{4}$ different graphs. | 0.25 | 7,190.0625 | 4,278.25 | 8,160.666667 |
Consider the function \( g(x) = \sum_{k=3}^{12} (\lfloor kx \rfloor - k \lfloor x \rfloor) \) where \( \lfloor r \rfloor \) denotes the greatest integer less than or equal to \( r \). Determine how many distinct values \( g(x) \) can take for \( x \ge 0 \).
A) 42
B) 43
C) 44
D) 45
E) 46 | 45 | 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\... | 1 | 2,621.75 | 2,621.75 | -1 |
On an island, there are 1000 villages, each with 99 inhabitants. Each inhabitant is either a knight, who always tells the truth, or a liar, who always lies. It is known that the island has exactly 54,054 knights. One day, each inhabitant was asked the question: "Are there more knights or liars in your village?" It turn... | 638 | 0.4375 | 6,884.5 | 5,234.428571 | 8,167.888889 | |
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to the real number $x$, such as $\lfloor 3.2 \rfloor = 3$, $\lfloor -4.5 \rfloor = -5$. The area of the shape formed by points $(x, y)$ on the plane that satisfy $\lfloor x \rfloor^2 + \lfloor y \rfloor^2 = 50$ is \_\_\_\_\_\_. | 12 | 0.75 | 5,155.875 | 4,591.916667 | 6,847.75 | |
In rectangle $ABCD$, $P$ is a point on $BC$ so that $\angle APD=90^{\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown. $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. Find $SD$. (Express your answer as a common fracti... | \dfrac{28}{3} | 0.5625 | 6,194.9375 | 4,641.666667 | 8,192 | |
Given $sin(\alpha-\beta)=\frac{1}{3}$ and $cos\alpha sin\beta=\frac{1}{6}$, find $\cos \left(2\alpha +2\beta \right)$. | \frac{1}{9} | 0.75 | 5,354 | 4,992.083333 | 6,439.75 | |
A whole number was increased by 2, and its square decreased by 2016. What was the number initially (before the increase)? | -505 | 0.0625 | 7,798.5 | 2,232 | 8,169.6 | |
Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time? | \frac{1}{2} | 1. **Calculate the initial lap time**:
When Elisa started swimming, she completed 10 laps in 25 minutes. To find the time it took for one lap, we divide the total time by the number of laps:
\[
\text{Initial lap time} = \frac{25 \text{ minutes}}{10 \text{ laps}} = 2.5 \text{ minutes per lap}
\]
2. **Calcu... | 0.6875 | 419 | 418.272727 | 420.6 |
Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle? | 1/4 | 0.3125 | 7,654.5 | 6,536.4 | 8,162.727273 | |
A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank. | 4 - 2\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
In a city, from 7:00 to 8:00, is a peak traffic period, during which all vehicles travel at half their normal speed. Every morning at 6:50, two people, A and B, start from points A and B respectively and travel towards each other. They meet at a point 24 kilometers from point A. If person A departs 20 minutes later, th... | 48 | 0.125 | 7,582.5 | 5,020.5 | 7,948.5 | |
In triangle \(ABC\), angle \(C\) is \(60^\circ\) and the radius of the circumcircle of this triangle is \(2\sqrt{3}\).
A point \(D\) is taken on the side \(AB\) such that \(AD = 2DB\) and \(CD = 2\sqrt{2}\). Find the area of triangle \(ABC\). | 3\sqrt{2} | 0.0625 | 8,067.0625 | 6,193 | 8,192 | |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 7$. If $YZ = 100$, then what is $XY$? | 70\sqrt{2} | 1 | 2,228.125 | 2,228.125 | -1 | |
Complex numbers $p$, $q$, and $r$ are zeros of a polynomial $Q(z) = z^4 - 2z^3 + sz + t,$ where $s, t \in \mathbb{C}$. It is given that $|p|^2 + |q|^2 + |r|^2 = 300$. The points corresponding to $p$, $q$, and $r$ in the complex plane form an equilateral triangle. Find the square of the side length of this triangle, den... | 225 | 0 | 8,179.75 | -1 | 8,179.75 | |
If $f(n) = n^2 + n + 17$, what is the value of $f(11)$? | 149 | 0.9375 | 2,259.125 | 1,863.6 | 8,192 | |
The product of the midline of a trapezoid and the segment connecting the midpoints of its diagonals equals 25. Find the area of the trapezoid if its height is three times the difference of its bases. | 150 | 1 | 2,959.375 | 2,959.375 | -1 | |
What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$? | -37 | 1 | 2,400.75 | 2,400.75 | -1 | |
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains.... | 10 | 0 | 8,192 | -1 | 8,192 | |
For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$? | 2 | 1 | 1,483.25 | 1,483.25 | -1 | |
Given a circle is inscribed in a triangle with side lengths $9, 12,$ and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$. | \frac{1}{2} | 0.25 | 5,226.75 | 4,794.5 | 5,370.833333 | |
Our club has 10 members, and wishes to pick a president, secretary, treasurer, and morale officer. In how many ways can we choose the officers, if individual members can only hold at most one office? | 5,\!040 | 0 | 1,508.1875 | -1 | 1,508.1875 | |
Amelia has a coin that lands heads with a probability of $\frac{3}{7}$, and Blaine has a coin that lands on heads with a probability of $\frac{1}{4}$. Initially, they simultaneously toss their coins once. If both get heads, they stop; otherwise, if either gets a head and the other a tail, that person wins. If both get ... | \frac{9}{14} | 0 | 8,192 | -1 | 8,192 | |
There exist positive integers $a,$ $b,$ and $c$ such that
\[3 \sqrt{\sqrt[3]{5} - \sqrt[3]{4}} = \sqrt[3]{a} + \sqrt[3]{b} - \sqrt[3]{c}.\]Find $a + b + c.$ | 47 | 0 | 8,192 | -1 | 8,192 | |
In 2010, the ages of a brother and sister were 16 and 10 years old, respectively. In what year was the brother's age twice that of the sister's? | 2006 | 1 | 600.0625 | 600.0625 | -1 | |
8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them? | \dfrac{37}{256} | 1 | 3,845.375 | 3,845.375 | -1 | |
Given the line $x+2y=a$ intersects the circle $x^2+y^2=4$ at points $A$ and $B$, and $|\vec{OA}+ \vec{OB}|=|\vec{OA}- \vec{OB}|$, where $O$ is the origin, determine the value of the real number $a$. | -\sqrt{10} | 0 | 6,455.625 | -1 | 6,455.625 | |
For a positive real number $x > 1,$ the Riemann zeta function $\zeta(x)$ is defined by
\[\zeta(x) = \sum_{n = 1}^\infty \frac{1}{n^x}.\]Compute
\[\sum_{k = 2}^\infty \{\zeta(2k - 1)\}.\]Note: For a real number $x,$ $\{x\}$ denotes the fractional part of $x.$ | \frac{1}{4} | 0.4375 | 7,051.75 | 6,402 | 7,557.111111 | |
If $M = 2007 \div 3$, $N = M \div 3$, and $X = M - N$, then what is the value of $X$? | 446 | 1 | 355.0625 | 355.0625 | -1 | |
If \( x = \frac{2}{3} \) and \( y = \frac{3}{2} \), find the value of \( \frac{1}{3}x^8y^9 \). | \frac{1}{2} | 0.875 | 1,951.3125 | 2,123.142857 | 748.5 | |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | 24 | To solve this problem, we need to find the number of ways to schedule 3 mathematics courses (algebra, geometry, and number theory) in a 6-period day such that no two mathematics courses are taken in consecutive periods.
#### Step 1: Total ways to schedule without restrictions
First, we calculate the total number of wa... | 0.8125 | 6,564.5625 | 6,189 | 8,192 |
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