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Let \( n \) be a positive integer. If
\[ n \equiv r \pmod{2} \ (r \in \{0,1\}), \]
find the number of integer solutions to the system of equations
\[ \begin{cases}
x + y + z = r, \\
|x| + |y| + |z| = n.
\end{cases} \] | 3n | 1/8 |
A box contains 5 white balls and 6 black balls. You draw them out of the box, one at a time. What is the probability that the first four draws alternate in colors, starting with a black ball? | \frac{2}{33} | 1/8 |
Find all the real solutions to
\[\frac{(x - 1)(x - 2)(x - 3)(x - 4)(x - 3)(x - 2)(x - 1)}{(x - 2)(x - 4)(x - 2)} = 1.\]Enter all the solutions, separated by commas. | 2 + \sqrt{2}, 2 - \sqrt{2} | 1/8 |
(a) Prove that for all \(a, b, c, d \in \mathbb{R}\) with \(a+b+c+d=0\),
\[ \max (a, b) + \max (a, c) + \max (a, d) + \max (b, c) + \max (b, d) + \max (c, d) \geq 0 \]
(b) Find the largest non-negative integer \(k\) such that it is possible to replace \(k\) of the six maxima in this inequality by minima in such a way that the inequality still holds for all \(a, b, c, d \in \mathbb{R}\) with \(a+b+c+d=0\). | 2 | 1/8 |
The shaded region shown consists of 11 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $x$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\pi$.
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[/asy] | 37\pi | 7/8 |
Solve the equation
\[ x^3 = 3y^3 + 9z^3 \]
in non-negative integers \(x\), \(y\), and \(z\). | (0,0,0) | 4/8 |
How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. | 248 | 2/8 |
Given the function $f(x) = \sqrt{\log_{3}(4x-1)} + \sqrt{16-2^{x}}$, its domain is A.
(1) Find the set A;
(2) If the function $g(x) = (\log_{2}x)^{2} - 2\log_{2}x - 1$, and $x \in A$, find the maximum and minimum values of the function $g(x)$ and the corresponding values of $x$. | -2 | 3/8 |
The first term of an arithmetic sequence is $a_{1}$, and the sum of its first and $n$-th term is $s_{a}$. Another arithmetic sequence has the corresponding data $b_{1}$ and $s_{b}$. A third sequence is formed in such a way that the term of arbitrary position is equal to the product of the terms of the same position from the first two sequences. What is the sum of the first $n$ terms of the third sequence? | \frac{n}{6(n-1)}[(2n-1)s_as_b-(n+1)(a_1s_b+b_1s_a-2a_1b_1)] | 1/8 |
Compute $\frac{x^8 + 16x^4 + 64 + 4x^2}{x^4 + 8}$ when $x = 3$. | 89 + \frac{36}{89} | 1/8 |
In terms of \( n \), what is the minimum number of edges a finite graph with chromatic number \( n \) could have? Prove your answer. | \frac{n(n-1)}{2} | 1/8 |
Given 10 distinct points on a plane, consider the midpoints of all segments connecting all pairs of points. What is the minimum number of such midpoints that could result? | 17 | 1/8 |
There are $n$ mathematicians attending a conference. Each mathematician has exactly 3 friends (friendship is mutual). If they are seated around a circular table such that each person has their friends sitting next to them on both sides, the number of people at the table is at least 7. Find the minimum possible value of $n$. | 24 | 3/8 |
Prove that the square of the distance between the centroid of a triangle and the incenter is equal to $\frac{1}{9}\left(p^{2}+5 r^{2}-16 R r\right)$. | \frac{1}{9}(p^2+5r^2-16Rr) | 1/8 |
Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations
\[
\begin{cases}
\gcd(a, b, c) = 22 \\
\mathrm{lcm}(a, b, c) = 2^{16} \cdot 11^{19}
\end{cases}
\] | 9720 | 7/8 |
Suppose $n \ge 0$ is an integer and all the roots of $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$ . | -3 | 4/8 |
In triangle \(ABC\), a line \(DE\) is drawn parallel to the base \(AC\). The area of triangle \(ABC\) is 8 square units, and the area of triangle \(DEC\) is 2 square units. Find the ratio of the length of segment \(DE\) to the length of the base of triangle \(ABC\). | 1:2 | 1/8 |
The base of a right triangular prism is an isosceles triangle \( ABC \) with \( AB = BC = a \) and \(\angle BAC = \alpha\). A plane is drawn through the side \( AC \) at an angle \(\varphi(\varphi<\pi / 2)\) to the base. Find the area of the cross-section, given that the cross-section is a triangle. | \frac{^2\sin2\alpha}{2\cos\varphi} | 2/8 |
Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right, and then reversing the sequence. For example, $b_4 = 43211234$ and $b_{12} = 121110987654321123456789101112$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9? | 22 | 2/8 |
Let \( a_{1}, a_{2}, \cdots \) be a sequence of positive real numbers such that for all \( i, j = 1, 2, \cdots \), the inequality \( a_{i+j} \leq a_{i} + a_{j} \) holds. Prove that for any positive integer \( n \), the sum \( a_{1}+\frac{a_{2}}{2}+\frac{a_{3}}{3}+\cdots +\frac{a_{n}}{n} \geq a_{n} \). | a_{1}+\frac{a_{2}}{2}+\frac{a_{3}}{3}+\cdots+\frac{a_{n}}{n}\gea_{n} | 1/8 |
In a right prism with triangular bases, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base) is 24, find the maximum volume of the prism.
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unitsize(1 cm);
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[/asy] | 16 | 7/8 |
Given that the general term of the sequence $\{a_n\}$ is $a_n=2^{n-1}$, and the general term of the sequence $\{b_n\}$ is $b_n=3n$, let set $A=\{a_1,a_2,\ldots,a_n,\ldots\}$, $B=\{b_1,b_2,\ldots,b_n,\ldots\}$, $n\in\mathbb{N}^*$. The sequence $\{c_n\}$ is formed by arranging the elements of set $A\cup B$ in ascending order. Find the sum of the first 28 terms of the sequence $\{c_n\}$, denoted as $S_{28}$. | 820 | 4/8 |
The sum of \( m \) distinct positive even numbers and \( n \) distinct positive odd numbers is 1987. For all such \( m \) and \( n\), what is the maximum value of \( 3m + 4n \)? Please prove your conclusion. | 221 | 1/8 |
In an exam, 153 people scored no more than 30 points, with an average score of 24 points. 59 people scored no less than 80 points, with an average score of 92 points. The average score of those who scored more than 30 points is 62 points. The average score of those who scored less than 80 points is 54 points. How many people participated in this exam? | 1007 | 5/8 |
In a sequence, the first term is \(a_1 = 2010\) and the second term is \(a_2 = 2011\). The values of the other terms satisfy the relation:
\[ a_n + a_{n+1} + a_{n+2} = n \]
for all \(n \geq 1\). Determine \(a_{500}\). | 2177 | 4/8 |
Petya and Vasya participated in the elections for the position of president of the chess club. By noon, Petya had 25% of the votes, and Vasya had 45%. After noon, only Petya's friends came to vote (and accordingly, only voted for him). As a result, Vasya ended up with only 27% of the votes. What percentage of the votes did Petya receive? | 55 | 6/8 |
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?
$\textbf{(A) } \frac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}$ | \textbf{(E)}\frac{4}{3}-\frac{4\sqrt{3}\pi}{27} | 1/8 |
The diagram shows a smaller rectangle made from three squares, each of area \(25 \ \mathrm{cm}^{2}\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in \(\mathrm{cm}^{2}\), of the larger rectangle? | 150 | 1/8 |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(0,3)$, the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______. | \frac{6\sqrt{5}}{5} | 6/8 |
If $183a8$ is a multiple of 287, find $a$.
The number of positive factors of $a^{2}$ is $b$, find $b$.
In an urn, there are $c$ balls, $b$ of them are either black or red, $(b+2)$ of them are either red or white, and 12 of them are either black or white. Find $c$.
Given $f(3+x)=f(3-x)$ for all values of $x$, and the equation $f(x)=0$ has exactly $c$ distinct roots, find $d$, the sum of these roots. | 48 | 5/8 |
Given an equilateral triangle \(ABC\). Point \(K\) is the midpoint of side \(AB\), point \(M\) lies on side \(BC\) such that \(BM : MC = 1 : 3\). On side \(AC\), point \(P\) is chosen such that the perimeter of triangle \(PKM\) is minimized. In what ratio does point \(P\) divide side \(AC\)? | 2:3 | 6/8 |
Given that $a$, $b$, $c$, $d$ are the thousands, hundreds, tens, and units digits of a four-digit number, respectively, and the digits in lower positions are not less than those in higher positions. When $|a-b|+|b-c|+|c-d|+|d-a|$ reaches its maximum value, the minimum value of this four-digit number is ____. | 1119 | 7/8 |
Find a natural number \( A \) such that if \( A \) is appended to itself on the right, the resulting number is a perfect square. | 13223140496 | 1/8 |
Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but now the price of wheat has risen to 1.6 yuan per kilogram, so what he pays is equivalent to 80 yuan per mu of land. Through Li Qiang's calculations, you can find out how many mu of land this is. | 20 | 6/8 |
Given that Xiao Ming ran a lap on a 360-meter circular track at a speed of 5 meters per second in the first half of the time and 4 meters per second in the second half of the time, determine the time taken to run in the second half of the distance. | 44 | 5/8 |
Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees. | 133.\overline{3} | 1/8 |
Divide an $m$-by-$n$ rectangle into $m n$ nonoverlapping 1-by-1 squares. A polyomino of this rectangle is a subset of these unit squares such that for any two unit squares $S, T$ in the polyomino, either (1) $S$ and $T$ share an edge or (2) there exists a positive integer $n$ such that the polyomino contains unit squares $S_{1}, S_{2}, S_{3}, \ldots, S_{n}$ such that $S$ and $S_{1}$ share an edge, $S_{n}$ and $T$ share an edge, and for all positive integers $k<n, S_{k}$ and $S_{k+1}$ share an edge. We say a polyomino of a given rectangle spans the rectangle if for each of the four edges of the rectangle the polyomino contains a square whose edge lies on it. What is the minimum number of unit squares a polyomino can have if it spans a 128-by343 rectangle? | 470 | 5/8 |
Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$ , $b$ , and $c$ are real numbers.
[i]Proposed by Tristan Shin | 4 | 1/8 |
Find the number of ordered quintuples $(a,b,c,d,e)$ of nonnegative real numbers such that:
\begin{align*}
a^2 + b^2 + c^2 + d^2 + e^2 &= 5, \\
(a + b + c + d + e)(a^3 + b^3 + c^3 + d^3 + e^3) &= 25.
\end{align*} | 31 | 5/8 |
A palindrome is a number that reads the same from left to right and right to left. For example, the numbers 333 and 4884 are palindromes. It is known that a three-digit number \( x \) is a palindrome. When 22 is added to it, a four-digit number is obtained, which is also a palindrome. Find \( x \). | 979 | 3/8 |
What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$ ?
$\textbf{(A)} ~3 \qquad\textbf{(B)} ~2\sqrt{3} \qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~3\sqrt{3}\qquad\textbf{(E)} ~6$ | \textbf{(C)}~4 | 1/8 |
Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points. | 95.44\% | 7/8 |
If $2^a+2^b=3^c+3^d$, the number of integers $a,b,c,d$ which can possibly be negative, is, at most: | 0 | 1/8 |
Let \( S = \{1, 2, 3, \ldots, 65\} \). Find the number of 3-element subsets \(\{a_{1}, a_{2}, a_{3}\}\) of \( S \) such that \( a_{i} \leq a_{i+1} - (i+2) \) for \( i = 1, 2 \). | 34220 | 4/8 |
Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?
$\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24$ | \textbf{(E)24} | 1/8 |
1. Solve the trigonometric inequality: $\cos x \geq \frac{1}{2}$
2. In $\triangle ABC$, if $\sin A + \cos A = \frac{\sqrt{2}}{2}$, find the value of $\tan A$. | -2 - \sqrt{3} | 7/8 |
From a bottle containing 1 liter of alcohol, $\frac{1}{3}$ liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, $\frac{1}{3}$ liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly. Finally, 1 liter of the mixture is poured out and an equal amount of water is added. How much alcohol is left in the bottle? | \frac{8}{27} | 1/8 |
Line $l_{1}$ is parallel to line $l_{2}$. There are 5 distinct points on $l_{1}$ and 10 distinct points on $l_{2}$. Line segments are formed by connecting the points on $l_{1}$ to the points on $l_{2}$. If no three line segments intersect at the same point, then the total number of intersection points among these line segments is $\qquad$. (Provide a specific number as the answer) | 450 | 7/8 |
In a finite sequence of real numbers, the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence. | 16 | 1/8 |
There are 13 students in a class (one of them being the monitor) and 13 seats in the classroom. Every day, the 13 students line up in random order and then enter the classroom one by one. Except for the monitor, each student will randomly choose an unoccupied seat and sit down. The monitor, however, prefers the seat next to the door and chooses it if possible. What is the probability that the monitor can choose his favourite seat? | 7/13 | 2/8 |
Let $A B C$ be a triangle and $\omega$ be its circumcircle. The point $M$ is the midpoint of arc $B C$ not containing $A$ on $\omega$ and $D$ is chosen so that $D M$ is tangent to $\omega$ and is on the same side of $A M$ as $C$. It is given that $A M=A C$ and $\angle D M C=38^{\circ}$. Find the measure of angle $\angle A C B$. | 33^{\circ} | 2/8 |
Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g. | 2020 | 1/8 |
A fair die is rolled six times. The probability of rolling at least a five at least five times is | \frac{13}{729} | 7/8 |
The chord \( A B \) subtends an arc of the circle equal to \( 120^{\circ} \). Point \( C \) lies on this arc, and point \( D \) lies on the chord \( A B \). Additionally, \( A D = 2 \), \( B D = 1 \), and \( D C = \sqrt{2} \).
Find the area of triangle \( A B C \). | \frac{3 \sqrt{2}}{4} | 7/8 |
What nine-digit number, when multiplied by 123456789, will produce a product that has the digits 9, 8, 7, 6, 5, 4, 3, 2, 1 in its last nine places (in that exact order)? | 989010989 | 1/8 |
A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$ , and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$ . Find the smallest $x$ for which $f(x)=f(2001)$ . | 429 | 6/8 |
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$. | 7\sqrt{3} | 5/8 |
Find the sum of the smallest and largest possible values for \( x \) which satisfy the following equation.
\[ 9^{x+1} + 2187 = 3^{6x - x^2} \] | 5 | 1/8 |
Define $n_a!$ for $n$ and $a$ positive to be
\[n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)\]
where $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to
$\textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$ | \textbf{(D)}\4^9 | 1/8 |
A square fits snugly between a horizontal line and two touching circles with a radius of 1000. The line is tangent to the circles. What is the side length of the square? | 400 | 7/8 |
Let \(\mathbb{R}\) denote the set of all real numbers. Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that
\[ f\left(x^{3}\right) + f(y)^{3} + f(z)^{3} = 3 x y z \]
for all \(x, y, z \in \mathbb{R}\) such that \(x + y + z = 0\). | f(x)=x | 1/8 |
A 30 foot ladder is placed against a vertical wall of a building. The foot of the ladder is 11 feet from the base of the building. If the top of the ladder slips 6 feet, then the foot of the ladder will slide how many feet? | 9.49 | 3/8 |
A line passing through the center of the circumscribed circle and the intersection point of the altitudes of a scalene triangle \(ABC\) divides its perimeter and area in the same ratio. Find this ratio. | 1:1 | 1/8 |
Find all pairs of primes \((p, q)\) for which \(p - q\) and \(pq - q\) are both perfect squares. | (3,2) | 1/8 |
A square flag features a green cross of uniform width with a yellow square in the center on a black background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 50% of the area of the flag, what percent of the area of the flag is yellow? | 6.25\% | 1/8 |
There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty boxes. How many boxes were there in total? Answer: 28182. | 28182 | 6/8 |
Let $m=n^{4}+x$, where $n \in \mathbf{N}$ and $x$ is a two-digit positive integer. Which value of $x$ ensures that $m$ is always a composite number? | 64 | 1/8 |
Consider the acute angle $ABC$ . On the half-line $BC$ we consider the distinct points $P$ and $Q$ whose projections onto the line $AB$ are the points $M$ and $N$ . Knowing that $AP=AQ$ and $AM^2-AN^2=BN^2-BM^2$ , find the angle $ABC$ . | 45 | 7/8 |
Two $4 \times 4$ squares are randomly placed on an $8 \times 8$ chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap? | 529/625 | 1/8 |
It is given that \( a, b \), and \( c \) are three positive integers such that
\[ a^{2} + b^{2} + c^{2} = 2011. \]
Let the highest common factor (HCF) and the least common multiple (LCM) of the three numbers \( a, b, \) and \( c \) be denoted by \( x \) and \( y \) respectively. Suppose that \( x + y = 388 \). Find the value of \( a + b + c \). | 61 | 5/8 |
Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$ 0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$ \text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2. $$ Prove that $$ \sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}. $$ *Proposed by David Speyer and Kiran Kedlaya* | \frac{2}{\sqrt{n-1}} | 1/8 |
Given $f(x)= \frac{2x}{x+1}$, calculate the value of the expression $f\left( \frac{1}{2016}\right)+f\left( \frac{1}{2015}\right)+f\left( \frac{1}{2014}\right)+\ldots+f\left( \frac{1}{2}\right)+f(1)+f(2)+\ldots+f(2014)+f(2015)+f(2016)$. | 4031 | 7/8 |
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. | 336 | 1/8 |
Determine the number of pairs \((a, b)\) of integers with \(1 \leq b < a \leq 200\) such that the sum \((a+b) + (a-b) + ab + \frac{a}{b}\) is a square of a number. | 112 | 7/8 |
In how many ways can one arrange all the natural numbers from 1 to \(2n\) in a circle so that each number is a divisor of the sum of its two adjacent numbers? (Arrangements that differ by rotation and symmetry are considered the same.) | 1 | 1/8 |
Given a quadrilateral pyramid \( P-ABCD \) with a base \( ABCD \) that is a rhombus with side length 2 and \( \angle ABC = 60^\circ \). Point \( E \) is the midpoint of \( AB \). \( PA \) is perpendicular to the plane \( ABCD \), and the sine of the angle between \( PC \) and the plane \( PAD \) is \( \frac{\sqrt{6}}{4} \).
(1) Find a point \( F \) on edge \( PD \) such that \( AF \parallel \) plane \( PEC \);
(2) Find the cosine of the dihedral angle between planes \( D-PE-A \). | \frac{4\sqrt{31}}{31} | 2/8 |
How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$?
$\text{(A)}\ 13 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$ | (A)13 | 1/8 |
The convex quadrilateral $ABCD$ has area $1$ , and $AB$ is produced to $E$ , $BC$ to $F$ , $CD$ to $G$ and $DA$ to $H$ , such that $AB=BE$ , $BC=CF$ , $CD=DG$ and $DA=AH$ . Find the area of the quadrilateral $EFGH$ . | 5 | 4/8 |
Determine all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(1) > 0 \) and for all \( (m, n) \in \mathbb{N}^{2} \), we have
\[ f\left(m^{2}+n^{2}\right)=f(m)^{2}+f(n)^{2} \] | f(n)=n | 1/8 |
Given that the area of triangle $\triangle ABC$ is $S$, points $D$, $E$, and $F$ are the midpoints of $BC$, $CA$, and $AB$ respectively. $I_{1}$, $I_{2}$, and $I_{3}$ are the incenters of triangles $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ respectively. Determine the value of the ratio $S^{\prime} / S$, where $S^{\prime}$ is the area of triangle $\triangle I_{1}I_{2}I_{3}$. | \frac{1}{4} | 5/8 |
Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. | (3 + \sqrt{15})^3 | 1/8 |
Given a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \). | \frac{3}{4} | 6/8 |
Find all prime numbers \( p \) such that \( 8p^4 - 3003 \) is a positive prime. | 5 | 7/8 |
How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct? | 5040 | 6/8 |
In $\triangle ABC$, $B(-\sqrt{5}, 0)$, $C(\sqrt{5}, 0)$, and the sum of the lengths of the medians on sides $AB$ and $AC$ is $9$.
(Ⅰ) Find the equation of the trajectory of the centroid $G$ of $\triangle ABC$.
(Ⅱ) Let $P$ be any point on the trajectory found in (Ⅰ), find the minimum value of $\cos\angle BPC$. | -\frac{1}{9} | 5/8 |
The cubic function \( y = x^3 + ax^2 + bx + c \) intersects the x-axis at points \( A, T, \) and \( B \) in that order. Tangents to the cubic curve at points \( P \) and \( Q \) are drawn from points \( A \) and \( B \) respectively, where \( P \) does not coincide with \( A \) and \( Q \) does not coincide with \( B \). The ratio of the projections of vectors \( \overrightarrow{AB} \) and \( \overrightarrow{PQ} \) onto the x-axis is: | -2 | 1/8 |
On the coordinate plane, a parabola \( y = x^2 \) is drawn. A point \( A \) is taken on the positive half of the \( y \)-axis, and two lines with positive slopes are drawn through it. Let \( M_1, N_1 \) and \( M_2, N_2 \) be the points of intersection with the parabola of the first and second lines, respectively. Find the ordinate of point \( A \), given that \( \angle M_1 O N_1 = \angle M_2 O N_2 \), where \( O \) is the origin. | 1 | 5/8 |
Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than twice any other, then find the minimum value of the product $xyz.$ | \frac{1}{32} | 3/8 |
In a square $ABCD$ with side length $4$, find the probability that $\angle AMB$ is an acute angle. | 1-\dfrac{\pi}{8} | 7/8 |
There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \text{cm}^2$. What is the area (in $\text{cm}^2$) of the square inscribed in the same $\triangle ABC$ as shown in Figure 2 below?
[asy] draw((0,0)--(10,0)--(0,10)--cycle); draw((-25,0)--(-15,0)--(-25,10)--cycle); draw((-20,0)--(-20,5)--(-25,5)); draw((6.5,3.25)--(3.25,0)--(0,3.25)--(3.25,6.5)); label("A", (-25,10), W); label("B", (-25,0), W); label("C", (-15,0), E); label("Figure 1", (-20, -5)); label("Figure 2", (5, -5)); label("A", (0,10), W); label("B", (0,0), W); label("C", (10,0), E); [/asy]
$\textbf{(A)}\ 378 \qquad \textbf{(B)}\ 392 \qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 441 \qquad \textbf{(E)}\ 484$
| 392 | 1/8 |
Each face of a tetrahedron is a triangle with sides $a, b,$ c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$ . | 8 | 7/8 |
Yan is somewhere between his office and a concert hall. To get to the concert hall, he can either walk directly there, or walk to his office and then take a scooter to the concert hall. He rides 5 times as fast as he walks, and both choices take the same amount of time. What is the ratio of Yan's distance from his office to his distance from the concert hall? | \frac{2}{3} | 7/8 |
Bob sends a secret message to Alice using her RSA public key $n = 400000001.$ Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n.$ Eve knows that $n = pq,$ a product of two prime factors. Find $p$ and $q.$ | 20201 | 3/8 |
Let \(a, b, c\) be distinct real numbers. Prove that:
\[
\left(\frac{2a - b}{a - b}\right)^2 + \left(\frac{2b - c}{b - c}\right)^2 + \left(\frac{2c - a}{c - a}\right)^2 \geq 5.
\] | 5 | 5/8 |
In acute triangle \( \triangle ABC \), \( AD \) and \( BE \) are the altitudes on sides \( BC \) and \( AC \) respectively. Points \( F \) and \( G \) lie on segments \( AD \) and \( BE \) respectively, such that \( \frac{AF}{FD} = \frac{BG}{GE} \). Line \( CF \) intersects \( BE \) at point \( H \), and line \( CG \) intersects \( AD \) at point \( I \). Prove that the four points \( F \), \( G \), \( I \), and \( H \) are concyclic (lie on the same circle). | F,G,I,H | 1/8 |
In the regular quadrilateral pyramid $P-ABCD$, $G$ is the centroid of $\triangle PBC$. Find the value of $\frac{V_{G-PAD}}{V_{G-PAB}}$. | 2 | 7/8 |
Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P(1, \frac{\sqrt{2}}{2})$ is on the ellipse, and $|PF_1| + |PF_2| = 2\sqrt{2}$.
$(1)$ Find the standard equation of ellipse $C$;
$(2)$ A line $l$ passing through $F_2$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle AOB$. | \frac{\sqrt{2}}{2} | 1/8 |
Given that \( p \) is a prime number, prove that there exists a prime number \( q \) such that for any positive integer \( n \), \( q \nmid n^{p} - p \). | q | 6/8 |
In a convex hexagon, two diagonals are chosen independently at random. Find the probability that these diagonals intersect inside the hexagon (inside meaning not at a vertex). | \frac{5}{12} | 7/8 |
An isosceles trapezoid has bases of lengths \(a\) and \(b\) (\(a > b\)). A circle can be inscribed in this trapezoid. Find the distance between the centers of the inscribed and circumscribed circles of this trapezoid. | \frac{^2-b^2}{8\sqrt{}} | 3/8 |
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