The numbers $1, 2, \dots, 1999$ are written on the board. Two players take turn choosing $a,b$ from the board and erasing them then writing one of $ab$, $a+b$, $a-b$. The first player wants the last number on the board to be divisible by $1999$, the second player want to stop him. Determine the winner.

Let $x_1,x_2,\ldots ,x_n$ be real numbers, where $n\ge 2$, satisfying $\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1$ . For each $k$, find the maximal value of $|x_k|$.

A classroom has a row of $35$ coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least $1$ coat and at least $1$ empty hook. How many different numbers of coats can satisfy Paulina's pattern? $\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$\\ (need visuals)

Let $m,n$ be natural numbers. Prove that $$m^{m^{m^m}}+n^{n^{n^n}}\geq m^{n^{n^n}}+ n^{m^{m^m}}$$ [i](nooonuii)[/i]

A circle of radius $4$ is inscribed in a triangle $ABC$. We call $D$ the touchpoint between the circle and side BC. Let $CD =8$, $DB= 10$. What is the length of the sides $AB$ and $AC$?

Complex numbers $\omega_1, \ldots, \omega_n$ each have magnitude $1.$ Let $z$ be a complex number distinct from $\omega_1, \ldots, \omega_n$ such that $$\frac{z+\omega_1}{z-\omega_1}+\ldots+\frac{z+\omega_n}{z-\omega_n}=0.$$ Prove that $|z|=1.$

The management of a secret society is made up of $4$ people. To admit new partners they use the following criteria: $\bullet$ Only the $4$ members of the directory vote, being able to do it in $3$ ways: in favor, against or abstaining. $\bullet$ Each aspiring partner must obtain at least $2$ votes in favor and none against. At the last management meeting, $8$ requests for admission were examined. Of the total votes cast, there were $23$ votes in favor, $2$ votes against and $7$ abstaining. What is the highest and what is the lowest value that the number of approved admission requests can have on that occasion?

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$. Adam says, "$n$ leaves a remainder of $2$ when divided by $3$." Bendeguz says, "For some $k$, $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$. Then $2n - s = 20$." Dennis says, "For some $m$, if I have $m$ marbles, there are $n$ ways to choose two of them." If exactly one of them is lying, what is $n$?

In the set $A$ with $n$ elements, $[\sqrt{2n}]+2$ subsets are chosen such that the union of any three of them is equal to $A$. Prove that the union of any two of them is equal to $A$ as well.

Let $K$ commutative field with $8$ elements. Prove that $(\exists)a\in K$ such that $a^3=a+1$. [i]Mircea Becheanu[/i]

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?

We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties: 1. \( T \) has at least two distinct elements; 2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \). For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \). What is the largest possible number of elements that a [b]kawaii [/b]subset can have?

Connect the commom points of circle$x^2+(y-1)^2=1$ and ellipse $9x^2+(y+1)^2=9$ with line segments, the figure is a $\text{(A)}$ line segment $\text{(B)}$ scalene triangle $\text{(C)}$ equilateral triangle $\text{(D)}$ quadrilateral

A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments as shown below. This cup has height $h$ and volume $V$. A second cup that is exactly the same shape as the first is held upright inside the first cup so that its base is a distance of $\tfrac{h}2$ from the base of the first cup. The volume of liquid that will t inside the first cup and outside the second cup can be written $\tfrac{m}{n}\cdot V$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] pair s = (10,1); draw(ellipse((0,0),4,1)^^ellipse((0,-6),2,.5)); fill((3,-6)--(-3,-6)--(0,-2.1)--cycle,white); draw((4,0)--(2,-6)^^(-4,0)--(-2,-6)); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-2.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); pair s = (10,-2); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-4.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); //darn :([/asy]

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?

Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves. [i]Palmer Mebane[/i]

A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle. [color=red][Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.][/color]

A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was $ \textbf{(A)}\ 54 \text{ or } 18 \qquad \textbf{(B)}\ 60 \text{ or } 15 \qquad \textbf{(C)}\ 63 \text{ or } 12 \qquad \textbf{(D)}\ 72 \text{ or } 36 \qquad \textbf{(E)}\ 75 \text{ or } 20$

The diagram below shows four congruent squares and some of their diagonals. Let $T$ be the number of triangles and $R$ be the number of rectangles that appear in the diagram. Find $T + R$. [img]https://cdn.artofproblemsolving.com/attachments/1/5/f756bbe67c09c19e811011cb6b18d0ff44be8b.png[/img]

Denote by $S$ the set of all primes $p$ such that the decimal representation of $\frac{1}{p}$ has the fundamental period of divisible by $3$. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}= 0.a_{1}a_{2}\cdots a_{3r}a_{1}a_{2}\cdots a_{3r}\cdots,\] where $r=r(p)$. For every $p \in S$ and every integer $k \ge 1$ define \[f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.\] [list=a] [*] Prove that $S$ is finite. [*] Find the highest value of $f(k, p)$ for $k \ge 1$ and $p \in S$.[/list]

Evaluate \[\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)\]

Prove $ \ \ \ \frac{22}{7}\minus{}\pi \equal{}\int_0^1 \frac{x^4(1\minus{}x)^4}{1\plus{}x^2}\ dx$.