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 |
|---|---|---|---|---|---|---|
Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)? | 47/32 | 0 | 8,192 | -1 | 8,192 | |
Given that the random variable X follows a normal distribution N(2, σ²) and P(X≤4)=0.88, find P(0<X<4). | 0.76 | 0.4375 | 6,954 | 5,676.857143 | 7,947.333333 | |
In the Cartesian coordinate system, the equation of circle C is $x^2 + y^2 - 4x = 0$, and its center is point C. Consider the polar coordinate system with the origin as the pole and the non-negative half of the x-axis as the polar axis. Curve $C_1: \rho = -4\sqrt{3}\sin\theta$ intersects circle C at points A and B.
(1)... | 1:2 | 0.625 | 5,980.125 | 5,645.4 | 6,538 | |
Let $A B C$ be a triangle whose incircle has center $I$ and is tangent to $\overline{B C}, \overline{C A}, \overline{A B}$, at $D, E, F$. Denote by $X$ the midpoint of major arc $\widehat{B A C}$ of the circumcircle of $A B C$. Suppose $P$ is a point on line $X I$ such that $\overline{D P} \perp \overline{E F}$. Given ... | \frac{4 \sqrt{5}}{5} | Let $H$ be the orthocenter of triangle $D E F$. We claim that $P$ is the midpoint of $\overline{D H}$. Indeed, consider an inversion at the incircle of $A B C$, denoting the inverse of a point with an asterik. It maps $A B C$ to the nine-point circle of $\triangle D E F$. According to $\angle I A X=90^{\circ}$, we have... | 0 | 8,192 | -1 | 8,192 |
The number 2015 is split into 12 terms, and then all the numbers that can be obtained by adding some of these terms (from one to nine) are listed. What is the minimum number of numbers that could have been listed? | 10 | 0 | 8,192 | -1 | 8,192 | |
In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle? [asy]
size(100);
draw((0,0)--(8,0)--(4,4*sqrt(3))--cycle);
label("8 m",(4,0),S);
draw((4,-.2)--(4,.2));
draw((1.8,3.5)--(2.2,3.3));
draw((6.3,3.5)--(5.8,3.3));
[/asy] | 24 | 1 | 1,044.0625 | 1,044.0625 | -1 | |
In trapezoid $ABCD$ the lengths of the bases $AB$ and $CD$ are 8 and 17 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction. | \frac{64}{225} | 0.9375 | 4,217.4375 | 3,952.466667 | 8,192 | |
Ann made a $3$-step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$-step staircase?
[asy]
size(150);
defaultpen(linewidth(0.8));
path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
for(int i=0;i<=2;i=i+1) {
for(int j=0;j<=3-i;j=j+1) {... | 22 | To solve this problem, we first need to understand the pattern in the number of toothpicks required to build each step of the staircase.
1. **Identify the pattern:**
- A $1$-step staircase requires $4$ toothpicks.
- A $2$-step staircase requires $10$ toothpicks.
- A $3$-step staircase requires $18$ toothpicks... | 0.25 | 7,294.375 | 5,763 | 7,804.833333 |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | -2 | 1 | 1,505.6875 | 1,505.6875 | -1 | |
Last year, the East Sea Crystal City World's business hall made a profit of 3 million yuan. At the beginning of this year, it relocated to the new Crystal City business hall, expanding its scope of operations. To achieve higher profits, it is necessary to increase advertising efforts. It is expected that starting from ... | 52 | 0 | 6,717.875 | -1 | 6,717.875 | |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C_{1}$ are given by $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). Point $M$ moves on curve $C_{1}$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$. The trajectory of poi... | 2 \sqrt{3} | 0.75 | 5,104 | 5,162.166667 | 4,929.5 | |
Given 5 points \( A, B, C, D, E \) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment? | 135 | 0 | 3,682.25 | -1 | 3,682.25 | |
In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3\cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the ... | 10 | Let the coordinates of A, B and C be (-a, 0), (a, 0) and (0, h) respectively. Then $D = (\frac{3a}{4}, \frac{h}{4})$ and $E = (-\frac{a}{8},\frac{h}{8}).$ $EC^2 = 7$ implies $a^2 + 49h^2 = 448$; $EB^2 = 9$ implies $81a^2 + h^2 = 576.$ Solve this system of equations simultaneously, $a=\sqrt{7}$ and $h=3$. Area of the tr... | 0.8125 | 5,390 | 4,976.307692 | 7,182.666667 |
Find the greatest root of $f(x) = 15x^4-13x^2+2$. | \frac{\sqrt{6}}{3} | 0 | 3,051.125 | -1 | 3,051.125 | |
Let \( m \in \mathbb{N} \), and let the integer part of \( \log_2 m \) be denoted as \( f(m) \). Calculate the value of \( f(1) + f(2) + \cdots + f(1024) \). | 8204 | 0.5625 | 6,693.1875 | 5,919 | 7,688.571429 | |
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and in... | 129 | Let $P$ and $Q$ be the centers of circles $\mathcal{P}$ and $\mathcal{Q}$ , respectively.
Let $M$ be midpoint $BC, \beta = \angle ACB.$
Upper diagram shows that
$\sin 2\beta = \frac {4}{5}$ and $AC = 2 AB.$ Therefore $\cos 2\beta = \frac {3}{5}.$
Let $CH\perp l, BH'\perp l.$ Lower diagram shows that
$\angle CAE = \a... | 0.0625 | 8,027.5 | 5,560 | 8,192 |
We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the othe... | 120 | 0.25 | 7,201.1875 | 5,951.5 | 7,617.75 | |
To prevent the spread of the novel coronavirus, individuals need to maintain a safe distance of at least one meter between each other. In a certain meeting room with four rows and four columns of seats, the distance between adjacent seats is more than one meter. During the epidemic, for added safety, it is stipulated t... | 11 | 0 | 7,811.9375 | -1 | 7,811.9375 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\cos B$ in terms of $a$ and $r$. | \dfrac {3}{4} | 0.8125 | 4,602 | 4,217.615385 | 6,267.666667 | |
Given $x_{1}=4$, $x_{2}=5$, $x_{3}=6$, calculate the standard deviation of this sample. | \frac{\sqrt{6}}{3} | 0 | 1,833.3125 | -1 | 1,833.3125 | |
In a 2-dimensional Cartesian coordinate system, there are 16 lattice points \((i, j)\) where \(0 \leq i \leq 3\) and \(0 \leq j \leq 3\). If \(n\) points are selected from these 16 points, determine the minimum value of \(n\) such that there always exist four points which are the vertices of a square. | 11 | 0 | 8,113.9375 | -1 | 8,113.9375 | |
Karlsson eats three jars of jam and one jar of honey in 25 minutes, while Little Brother does it in 55 minutes. Karlsson eats one jar of jam and three jars of honey in 35 minutes, while Little Brother does it in 1 hour 25 minutes. How long will it take them to eat six jars of jam together? | 20 | 0.25 | 7,017.625 | 3,900.75 | 8,056.583333 | |
Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the "base" $-n+i$ using the integers $1,2,\ldots,n^2$ as "digits." That is, the equation\[r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0\]is true for a uni... | 490 | 0.8125 | 6,499.75 | 6,372.153846 | 7,052.666667 | |
Find all ordered pairs $(a, b)$ of complex numbers with $a^{2}+b^{2} \neq 0, a+\frac{10b}{a^{2}+b^{2}}=5$, and $b+\frac{10a}{a^{2}+b^{2}}=4$. | (1,2),(4,2),\left(\frac{5}{2}, 2 \pm \frac{3}{2} i\right) | First, it is easy to see that $ab \neq 0$. Thus, we can write $\frac{5-a}{b}=\frac{4-b}{a}=\frac{10}{a^{2}+b^{2}}$. Then, we have $\frac{10}{a^{2}+b^{2}}=\frac{4a-ab}{a^{2}}=\frac{5b-ab}{b^{2}}=\frac{4a+5b-2ab}{a^{2}+b^{2}}$. Therefore, $4a+5b-2ab=10$, so $(2a-5)(b-2)=0$. Now we just plug back in and get the four solut... | 0 | 8,191.125 | -1 | 8,191.125 |
Let \( A B C \) be a triangle. The midpoints of the sides \( B C, A C \), and \( A B \) are denoted by \( D, E \), and \( F \) respectively.
The two medians \( A D \) and \( B E \) are perpendicular to each other and have lengths \( \overline{A D} = 18 \) and \( \overline{B E} = 13.5 \).
Calculate the length of the t... | 22.5 | 0.375 | 6,453.8125 | 4,852.666667 | 7,414.5 | |
Let $C_1$ and $C_2$ be circles of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both $C_1$ and $C_2$? | 6 | 1. **Identify the configuration of $C_1$ and $C_2$:**
- $C_1$ and $C_2$ are circles of radius 1, tangent to each other. Let's denote the point of tangency as $P$. The distance between the centers of $C_1$ and $C_2$ is thus 2.
2. **Consider circles of radius 3 tangent to both $C_1$ and $C_2$:**
- **Externally tan... | 0.125 | 7,361.0625 | 6,852.5 | 7,433.714286 |
Two lines with slopes $-\frac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | 22.5 | 0 | 4,876.6875 | -1 | 4,876.6875 | |
Given the sequence $\{a_n\}$ satisfying $(\log_3{a_n}+1=\log_3{a_{n+1}}\ (n\in \mathbb{N}^*)$, and $(a_2+a_4+a_6=9$, find the value of $(\log_{\frac{1}{3}}(a_5+a_7+a_9))$. | -5 | 0.75 | 4,059.0625 | 2,683.333333 | 8,186.25 | |
Two lines with slopes 3 and -1 intersect at the point $(3, 1)$. What is the area of the triangle enclosed by these two lines and the horizontal line $y = 8$?
- **A)** $\frac{25}{4}$
- **B)** $\frac{98}{3}$
- **C)** $\frac{50}{3}$
- **D)** 36
- **E)** $\frac{200}{9}$ | \frac{98}{3} | 0 | 3,683.875 | -1 | 3,683.875 | |
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations \[a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2\] has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Then $\rho^2$ can be expressed as a fraction $\frac {... | 7 | Notice that the given equation implies
$a^2 + y^2 = b^2 + x^2 = 2(ax + by)$
We have $2by \ge y^2$, so $2ax \le a^2 \implies x \le \frac {a}{2}$.
Then, notice $b^2 + x^2 = a^2 + y^2 \ge a^2$, so $b^2 \ge \frac {3}{4}a^2 \implies \rho^2 \le \frac {4}{3}$.
The solution $(a, b, x, y) = \left(1, \frac {\sqrt {3}}{2}, \fr... | 0.0625 | 8,191.875 | 8,190 | 8,192 |
Let $x$ be a positive integer such that $9x\equiv 1\pmod{25}$.
What is the remainder when $11+x$ is divided by $25$? | 0 | 1 | 1,871.0625 | 1,871.0625 | -1 | |
A television station is broadcasting 5 advertisements in a row, which include 3 different commercial advertisements and 2 different National Games promotional advertisements. The requirements are that the last advertisement must be one of the National Games promotional advertisements, and the two National Games adverts... | 36 | 0.0625 | 7,753 | 7,288 | 7,784 | |
In the arithmetic sequence $\{a_{n}\}$, if $\frac{{a}_{9}}{{a}_{8}}＜-1$, and its sum of the first $n$ terms $S_{n}$ has a minimum value, determine the minimum value of $n$ for which $S_{n} \gt 0$. | 16 | 0 | 8,192 | -1 | 8,192 | |
Given the parabola $C: x^{2}=8y$ and its focus $F$, the line $PQ$ and $MN$ intersect the parabola $C$ at points $P$, $Q$, and $M$, $N$, respectively. If the slopes of the lines $PQ$ and $MN$ are $k_{1}$ and $k_{2}$, and satisfy $\frac{1}{{k_1^2}}+\frac{4}{{k_2^2}}=1$, then the minimum value of $|PQ|+|MN|$ is ____. | 88 | 0 | 8,192 | -1 | 8,192 | |
Seven students shared the cost of a $\$26.00$ pizza. Each student paid either $\$3.71$ or $\$3.72$. How many students paid $\$3.72$? | 3 | We can rephrase the given information by saying that each of the seven students paid $\$3.71$ and some of the students paid an extra $\$0.01$. Since $7 \times \$3.71=\$25.97$ and the pizza cost $\$26.00$ in total, then the students who paid the extra $\$0.01$ each must make up the final $\$0.03$ of the cost of the pizz... | 1 | 2,547.5625 | 2,547.5625 | -1 |
Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares. She can move the pieces in two ways: If a token has a free square in front of it, then she can move this token one square to the right. If the square immediately to the right of a token is occupied... | 42 | We put a marker on $(i, j)$ when a token is on $i$th and $j$th square and $i>j$. When the token in front/behind moves one step forward to a blank square, move the marker rightward/upward one unit correspondingly. When a 'leapfrog' happens, the marker moves from $(x-1, x)$ to $(x, x+1)$. We can translate this movement i... | 0 | 8,132.75 | -1 | 8,132.75 |
Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possible values for $k$. | 15 | 1 | 1,855.75 | 1,855.75 | -1 | |
In a sanatorium, there are grandmothers and grandfathers, with grandfathers being twice as many as grandmothers. The average age of the grandmothers is 5 years more than the average age of the grandfathers, and the average age of all retirees is greater than 77 years but less than 78. Find the average age of the grandm... | 81 | 0.6875 | 3,544.375 | 2,583.363636 | 5,658.6 | |
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(3,-\sqrt{3})$, with $x\in[0,\pi]$.
$(1)$ If $\overrightarrow{a}\parallel\overrightarrow{b}$, find the value of $x$; $(2)$ Let $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$, find the maximum and minimum values of $f(x)$ and the corresponding... | -2\sqrt{3} | 0.4375 | 6,030.25 | 5,908.714286 | 6,124.777778 | |
What is the reciprocal of $\frac{1}{2}+\frac{2}{3}$? | \frac{6}{7} | 1. **Find the common denominator** for the fractions $\frac{1}{2}$ and $\frac{2}{3}$:
\[
\frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{2}{3} = \frac{4}{6}
\]
Here, the least common multiple of 2 and 3 is 6.
2. **Add the fractions**:
\[
\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = ... | 1 | 1,520.25 | 1,520.25 | -1 |
In how many different ways can a chess king move from square $e1$ to square $h5$, if it is only allowed to move one square to the right, upward, or diagonally right-upward? | 129 | 0.375 | 6,953.5625 | 6,004.166667 | 7,523.2 | |
In triangle \(ABC\), sides \(AB\) and \(BC\) are equal, \(AC = 2\), and \(\angle ACB = 30^\circ\). From vertex \(A\), the angle bisector \(AE\) and the median \(AD\) are drawn to the side \(BC\). Find the area of triangle \(ADE\). | \frac{2 \sqrt{3} - 3}{6} | 0 | 8,091.5 | -1 | 8,091.5 | |
You would like to provide airline service to the 10 cities in the nation of Schizophrenia, by instituting a certain number of two-way routes between cities. Unfortunately, the government is about to divide Schizophrenia into two warring countries of five cities each, and you don't know which cities will be in each new ... | 30 | Each city $C$ must be directly connected to at least 6 other cities, since otherwise the government could put $C$ in one country and all its connecting cities in the other country, and there would be no way out of $C$. This means that we have 6 routes for each of 10 cities, counted twice (since each route has two endpo... | 0 | 8,083.9375 | -1 | 8,083.9375 |
It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have? | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Given that $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, $\cos\left(\frac{\pi}{4}+\alpha\right) = \frac{1}{3}$, $\cos\left(\frac{\pi}{4}-\frac{\beta}{2}\right) = \frac{\sqrt{3}}{3}$,
find
$(1)$ the value of $\cos \alpha$;
$(2)$ the value of $\cos\left(\alpha+\frac{\beta}{2}\right)$. | \frac{5\sqrt{3}}{9} | 0 | 7,845.6875 | -1 | 7,845.6875 | |
Given the right triangles ABC and ABD, what is the length of segment BC, in units? [asy]
size(150);
pair A, B, C, D, X;
A=(0,0);
B=(0,12);
C=(-16,0);
D=(-35,0);
draw(A--B--D--A);
draw(B--C);
draw((0,1.5)--(-1.5,1.5)--(-1.5,0));
label("$37$", (B+D)/2, NW);
label("$19$", (C+D)/2, S);
label("$16$", (A+C)/2, S);
label("A",... | 20 | 0.75 | 5,011.25 | 3,951 | 8,192 | |
Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$? | 18 | 0.9375 | 4,309.625 | 4,050.8 | 8,192 | |
Given $f(\alpha) = \frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\tan(-\alpha+\pi)}{-\tan(-\alpha -\pi )\cos( \frac {\pi}{2}-\alpha )}$:
1. Simplify $f(\alpha)$.
2. If $\alpha$ is an angle in the third quadrant and $\cos(\alpha- \frac {3\pi}{2}) = \frac {1}{5}$, find the value of $f(\alpha)$. | \frac {2\sqrt{6}}{5} | 0 | 4,018.625 | -1 | 4,018.625 | |
The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? | \frac{1}{19} | 1. **Prime Factorization of $21!$:**
To find the prime factorization of $21!$, we need to consider the contributions of each prime number up to 21. We count the number of times each prime divides any of the numbers from 1 to 21. This can be done using the formula for the highest power of a prime $p$ dividing $n!$:... | 0.875 | 4,353.5625 | 3,816.785714 | 8,111 |
Through a point $P$ inside the $\triangle ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas.
If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: | \frac{1}{2} | To solve this problem, we need to find the distance from point $P$ to the base $AB$ of $\triangle ABC$, given that a line through $P$ parallel to $AB$ divides the triangle into two regions of equal area.
1. **Understanding the Triangle and the Line through $P$:**
- Since the line through $P$ is parallel to $AB$ and... | 0 | 6,496.5 | -1 | 6,496.5 |
I live on a very short street with 14 small family houses. The odd-numbered houses from 1 are on one side of the street, and the even-numbered houses from 2 are on the opposite side (e.g., 1 and 2 are opposite each other).
On one side of the street, all families have surnames that are colors, and on the other side, t... | 13 | 0 | 8,192 | -1 | 8,192 | |
For all positive integers $n$ less than $2002$, let
\begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if } n \text{ is divisible by } 13 \text{ and } 14;\\ 13, & \text{if } n \text{ is divisible by } 14 \text{ and } 11;\\ 14, & \text{if } n \text{ is divisible by } 11 \text{ and } 13;\\ 0, & \text{otherwise... | 448 | 1. **Identify the conditions for $a_n$:**
- $a_n = 11$ if $n$ is divisible by both 13 and 14. The least common multiple (LCM) of 13 and 14 is $\operatorname{lcm}(13, 14) = 182$.
- $a_n = 13$ if $n$ is divisible by both 14 and 11. The least common multiple of 14 and 11 is $\operatorname{lcm}(14, 11) = 154$.
- $... | 0.75 | 4,831.375 | 4,231.916667 | 6,629.75 |
Let \( n \geq 2 \) be a fixed integer. Find the smallest constant \( C \) such that for all non-negative reals \( x_1, x_2, \ldots, x_n \):
\[ \sum_{i < j} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4. \]
Determine when equality occurs. | \frac{1}{8} | 0.1875 | 8,102.5 | 7,714.666667 | 8,192 | |
Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy? | 7 |
Let $x$ be the number of $1$ pairs of socks, $y$ be the number of $3$ pairs of socks, and $z$ be the number of $4$ pairs of socks that Ralph bought. We are given the following system of equations based on the problem statement:
1. **Total number of pairs**: $x + y + z = 12$
2. **Total cost of socks**: $x + 3y + 4z = ... | 1 | 2,235.875 | 2,235.875 | -1 |
The six-digit number \( 2PQRST \) is multiplied by 3, and the result is the six-digit number \( PQRST2 \). What is the sum of the digits of the original number? | 27 | 0.9375 | 4,914.625 | 4,696.133333 | 8,192 | |
Two 6-sided dice, one red and one green, are rolled. What is the probability that the red die shows an odd number and the green die shows a number that is a perfect square? | \frac16 | 1 | 1,660.5 | 1,660.5 | -1 | |
Given that the populations of three communities $A$, $B$, and $C$ are 600, 1200, and 1500 residents respectively, and if 15 people are drawn from community $C$, determine the total number of people drawn in the sample. | 33 | 0.1875 | 4,024.5625 | 3,407.333333 | 4,167 | |
Mia and Jake ordered a pizza cut into 12 equally-sized slices. Mia wanted a plain pizza but Jake wanted pepperoni on one-third of the pizza. The cost of a plain pizza was $12, and the additional cost for pepperoni on part of the pizza was $3. Jake ate all the pepperoni slices and three plain slices. Mia ate the rest. E... | 2.5 | 0 | 4,529.4375 | -1 | 4,529.4375 | |
Two complementary angles are in a ratio of $3:2$. What is the measure, in degrees, of the smaller angle? | 36 | 1 | 1,216.25 | 1,216.25 | -1 | |
Each of the numbers $a_1,$ $a_2,$ $\dots,$ $a_{95}$ is $\pm 1.$ Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 95} a_i a_j.\] | 13 | 0.9375 | 5,533.875 | 5,356.666667 | 8,192 | |
Evaluate the volume of solid $T$ defined by the inequalities $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$. | \frac{32}{3} | 0.125 | 8,144 | 7,808 | 8,192 | |
At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $24.50$. How much would $8$ adult tickets and $6$ child tickets cost? | 38.50 | 1. **Define the variables:**
Let $x$ be the price of an adult ticket. Then, the price of a child ticket is $\frac{x}{2}$.
2. **Set up the equation based on the given information:**
The total cost for $5$ adult tickets and $4$ child tickets is $24.50. Therefore, we can write the equation:
\[
5x + 4\left(\fr... | 1 | 2,566.125 | 2,566.125 | -1 |
Given the curves $C_{1}: \begin{cases} x=2+\cos t \\ y=\sin t-1 \end{cases}$ (with $t$ as the parameter), and $C_{2}: \begin{cases} x=4\cos \alpha \\ y=\sin \alpha \end{cases}$ (with $\alpha$ as the parameter), in the polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the p... | \frac { \sqrt {2}-1}{ \sqrt {2}} | 0 | 4,594.8125 | -1 | 4,594.8125 | |
How many diagonals does a convex polygon with 23 sides have? | 230 | 1 | 1,528.5625 | 1,528.5625 | -1 | |
How much money should I invest at an annually compounded interest rate of $5\%$ so that I have $\$500,\!000$ in ten years? Express your answer as a dollar value rounded to the nearest cent. | \$306,\!956.63 | 0 | 8,107.875 | -1 | 8,107.875 | |
Fully simplify the following expression: $[(2+3+4+5)\div2] + [(2\cdot5+8)\div3]$. | 13 | 1 | 1,751.0625 | 1,751.0625 | -1 | |
In $\triangle ABC$, $2\sin ^{2} \frac{A}{2}= \sqrt{3}\sin A$, $\sin (B-C)=2\cos B\sin C$, find the value of $\frac{AC}{AB}$ . | \frac{1+\sqrt{13}}{2} | 0 | 6,969.1875 | -1 | 6,969.1875 | |
Santa Claus has 36 identical gifts distributed among 8 bags. Each bag contains at least 1 gift, and the number of gifts in each of the 8 bags is unique. From these bags, select some bags such that the total number of gifts in the selected bags can be evenly divided among 8 children, with each child receiving at least o... | 31 | 0.0625 | 7,850.5 | 5,949 | 7,977.266667 | |
Compute $26\times33+67\times26$. | 2600 | 1 | 1,677.3125 | 1,677.3125 | -1 | |
Solve the equation: $2\left(x-1\right)^{2}=x-1$. | \frac{3}{2} | 0.1875 | 707.125 | 664.333333 | 717 | |
Megan has lost Fatima's phone number. Megan knows that the first three digits are either 296 or 299. The remaining four digits are 0, 1, 6 and 7, but she isn't sure of the order of these digits. If Megan randomly dials a seven-digit number that meets these conditions, what is the probability that she dials Fatima's cor... | \frac{1}{48} | 1 | 1,009.5625 | 1,009.5625 | -1 | |
Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\sqrt{n}+11i$. Find $m+n$. | 230 | Notice that we must have \[\frac{f(f(z))-f(z)}{f(z)-z}=-\frac{f(f(z))-f(z)}{z-f(z)}\in i\mathbb R .\]However, $f(t)-t=t(t-20)$, so \begin{align*} \frac{f(f(z))-f(z)}{f(z)-z}&=\frac{(z^2-19z)(z^2-19z-20)}{z(z-20)}\\ &=\frac{z(z-19)(z-20)(z+1)}{z(z-20)}\\ &=(z-19)(z+1)\\ &=(z-9)^2-100. \end{align*} Then, the real part of... | 0 | 8,192 | -1 | 8,192 |
A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\pi$. Find $W$, as a common fraction. | \frac{284}{3} | 1 | 2,616.9375 | 2,616.9375 | -1 | |
The diagonals \(AC\) and \(BD\) of the symmetric trapezoid \(ABCD\) intersect at point \(O\). The area of triangle \(AOB\) is \(52 \, \text{m}^2\) and the area of triangle \(COD\) is \(117 \, \text{m}^2\). Calculate the area of the trapezoid. | 325 | 0.625 | 6,382.5625 | 5,530.3 | 7,803 | |
In the plane of equilateral triangle $PQR$, points $S$, $T$, and $U$ are such that triangle $PQS$, $QRT$, and $RUP$ are also equilateral triangles. Given the side length of $PQR$ is 4 units, find the area of hexagon $PQURTS$. | 16\sqrt{3} | 0 | 7,356.6875 | -1 | 7,356.6875 | |
Circle $C$ has its center at $C(5, 5)$ and has a radius of 3 units. Circle $D$ has its center at $D(14, 5)$ and has a radius of 3 units. What is the area of the gray region bound by the circles and the $x$-axis?
```asy
import olympiad; size(150); defaultpen(linewidth(0.8));
xaxis(0,18,Ticks("%",1.0));
yaxis(0,9,Ticks("... | 45 - \frac{9\pi}{2} | 0 | 6,434.6875 | -1 | 6,434.6875 | |
When we say that Ray is climbing up the stairs $m$ at a time, we mean that he starts on the floor (step $0$) then jumps to step $m$ and then to $2m$ and so on until the number of steps to the top is less than $m$. Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $4$ steps at a time, there are ... | 27 | 1 | 2,548.1875 | 2,548.1875 | -1 | |
What is the probability of rolling six standard, six-sided dice and getting six distinct numbers? Express your answer as a common fraction. | \frac{5}{324} | 1 | 3,014.5625 | 3,014.5625 | -1 | |
Compute $$ \sum_{a_{1}=0}^{\infty} \sum_{a_{2}=0}^{\infty} \cdots \sum_{a_{7}=0}^{\infty} \frac{a_{1}+a_{2}+\cdots+a_{7}}{3^{a_{1}+a_{2}+\cdots+a_{7}}} $$ | 15309 / 256 | Note that, since this is symmetric in $a_{1}$ through $a_{7}$, $$ \sum_{a_{1}=0}^{\infty} \sum_{a_{2}=0}^{\infty} \cdots \sum_{a_{7}=0}^{\infty} \frac{a_{1}+a_{2}+\cdots+a_{7}}{3^{a_{1}+a_{2}+\cdots+a_{7}}} =7 \sum_{a_{1}=0}^{\infty} \sum_{a_{2}=0}^{\infty} \cdots \sum_{a_{7}=0}^{\infty} \frac{a_{1}}{3^{a_{1}+a_{2}+\cd... | 0.375 | 7,870.8125 | 7,335.5 | 8,192 |
Zeus starts at the origin \((0,0)\) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. What is the smallest number of moves that he can make to get to the point \((1056,1007)\)? | 2111 | 0 | 8,192 | -1 | 8,192 | |
A basketball team has 15 available players. Initially, 5 players start the game, and the other 10 are available as substitutes. The coach can make up to 4 substitutions during the game, under the same rules as the soccer game—no reentry for substituted players and each substitution is distinct. Calculate the number of ... | 51 | 0 | 6,339.8125 | -1 | 6,339.8125 | |
Given a triangle \(ABC\) with side lengths \(AB = BC = 80\) and \(AC = 96\).
Circle \(Q_1\) is inscribed in the triangle \(ABC\). Circle \(Q_2\) is tangent to \(Q_1\) and to the sides \(AB\) and \(BC\). Circle \(Q_3\) is tangent to \(Q_2\) and also to the sides \(AB\) and \(BC\). Find the radius of circle \(Q_3\). | 1.5 | 0 | 7,381.5625 | -1 | 7,381.5625 | |
Bill can buy jags, jigs and jogs for $\$1$, $\$2$ and $\$7$ each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $\$50$? | 6 | 0.5 | 6,073.5625 | 5,219.875 | 6,927.25 | |
Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$ | \frac{1}{2} | 0.3125 | 7,067.625 | 5,404 | 7,823.818182 | |
Simplify $(2^8+4^5)(2^3-(-2)^3)^{10}$. | 1342177280 | 0 | 3,408.875 | -1 | 3,408.875 | |
Given that 30 balls are put into four boxes A, B, C, and D, such that the sum of the number of balls in A and B is greater than the sum of the number of balls in C and D, find the total number of possible ways. | 2600 | 0.3125 | 7,029.0625 | 4,487 | 8,184.545455 | |
Real numbers $u$ and $v$ are each chosen independently and uniformly at random from the interval $(0, 2)$. What is the probability that $\lfloor \log_3 u \rfloor = \lfloor \log_3 v \rfloor$?
A) $\frac{1}{9}$
B) $\frac{1}{3}$
C) $\frac{4}{9}$
D) $\frac{5}{9}$
E) $\frac{1}{2}$ | \frac{5}{9} | 0 | 7,805.5 | -1 | 7,805.5 | |
Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients. | 259 | 1 | 3,342.3125 | 3,342.3125 | -1 | |
A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the ratio of the number of blue tiles to the number of white tiles, ignoring part tiles. | 1: 2 | 0 | 5,764.0625 | -1 | 5,764.0625 | |
A rectangular prism has a volume of $12 \mathrm{~cm}^{3}$. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. What is the volume of this new prism? | 144 | Suppose that the original prism has length $\ell \mathrm{cm}$, width $w \mathrm{~cm}$, and height $h \mathrm{~cm}$. Since the volume of this prism is $12 \mathrm{~cm}^{3}$, then $\ell w h=12$. The new prism has length $2 \ell \mathrm{cm}$, width $2 w \mathrm{~cm}$, and height 3 cm. The volume of this prism, in $\mathrm... | 0.25 | 820.375 | 1,517 | 588.166667 |
Let $n$ be the answer to this problem. Given $n>0$, find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$. | 48 | We explicitly compute the number of triangles satisfying the problem conditions for any $n$. There are three kinds of triangles: isosceles and scalene. (Equilateral triangles are isosceles.) - Case 1: Isosceles. A triangle with side lengths $a, a, b$ must satisfy $2a>b$ and $2a+b=n$. So $2a$ can be any even integer in ... | 0 | 8,192 | -1 | 8,192 |
Given a sequence where each term is either 1 or 2, starting with 1, and where between the \(k\)-th 1 and the \((k+1)\)-th 1 there are \(2^{k-1}\) 2's (i.e., the sequence is 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, ...), determine the sum of the first 1998 terms of this sequence. | 3986 | 0.0625 | 7,352.9375 | 6,956 | 7,379.4 | |
How many positive integer solutions does the equation have $$ \left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1? $$ ( $\lfloor x \rfloor$ denotes the integer part of $x$ , for example $\lfloor 2\rfloor = 2$ , $\lfloor \pi\rfloor = 3$ , $\lfloor \sqrt2 \rfloor =1$ ) | 110 | 0 | 8,075.75 | -1 | 8,075.75 | |
The square of an integer is called a perfect square. If $x$ is a perfect square, the next larger perfect square is | $x+2\sqrt{x}+1$ | 1. **Identify the nature of $x$**: Given that $x$ is a perfect square, we can express $x$ as $k^2$ for some integer $k$.
2. **Find the next perfect square**: The next integer after $k$ is $k+1$. The square of $k+1$ is:
\[
(k+1)^2 = k^2 + 2k + 1
\]
3. **Express in terms of $x$**: Since $x = k^2$, we substitut... | 0 | 3,566.5625 | -1 | 3,566.5625 |
Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$. | 064 | 0 | 5,978.625 | -1 | 5,978.625 | |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{7}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\] | -\frac{3}{4} | 0 | 8,043.75 | -1 | 8,043.75 | |
Given two distinct points $A, B$ and line $\ell$ that is not perpendicular to $A B$, what is the maximum possible number of points $P$ on $\ell$ such that $A B P$ is an isosceles triangle? | 5 | In an isosceles triangle, one vertex lies on the perpendicular bisector of the opposite side. Thus, either $P$ is the intersection of $A B$ and $\ell$, or $P$ lies on the circle centered at $A$ with radius $A B$, or $P$ lies on the circle centered at $B$ with radius $A B$. Each circle-line intersection has at most two ... | 0.1875 | 7,879.25 | 6,894 | 8,106.615385 |
From a container filled to the brim with $100\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water... | 1.75 | 0 | 8,192 | -1 | 8,192 | |
Consider all possible quadratic polynomials $x^2 + px + q$ with a positive discriminant, where the coefficients $p$ and $q$ are integers divisible by 5. Find the largest natural number $n$ such that for any polynomial with the described properties, the sum of the hundredth powers of the roots is an integer divisible b... | 50 | 0 | 8,151.875 | -1 | 8,151.875 | |
The three row sums and the three column sums of the array \[
\left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right]
\]
are the same. What is the least number of entries that must be altered to make all six sums different from one another? | 4 | 1. **Identify the initial sums**: First, we calculate the row sums and column sums of the given matrix:
\[
\begin{matrix}
4 & 9 & 2\\
8 & 1 & 6\\
3 & 5 & 7
\end{matrix}
\]
- Row sums: $4+9+2=15$, $8+1+6=15$, $3+5+7=15$
- Column sums: $4+8+3=15$, $9+1+5=15$, $2+6+7=15$
All sums are equal... | 0.0625 | 7,978.625 | 7,897 | 7,984.066667 |
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