Find all integers $n>1$ for which there exist positive integers $a_1,a_2,\ldots,a_n$ such that when divided by $a_i+a_j, 1\leq i\leq j\leq n$ there are $\frac{n(n+1)}{2}$ distinct remainders.

Find the area enclosed by the graph of $a^2x^4=b^2x^2-y^2\ (a>0,\ b>0).$

Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row $R_0$ in the board such that all numbers in $R_0$ are positive. Denote by $R_1$ the row below $R_0$ , by $R_2$ the row below $R_1$ etc. Show that for each $N \ge 1$ the row $R_N$ cannot contain more than $N$ zeroes.

Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ taken in such a way that $|AB| = 3 |AA_1| $ and $|AB| = 4 |A_2B|$, similarly consider points $B_1$ and $B_2, C_1$ and $C_2, D_1$ and $D_2$ respectively on the sides $BC$, $CD$ and $DA$. The intersection point of straight lines $D_2A_1$ and $A_2B_1$ is $E$, the intersection point of straight lines $A_2B_1$ and $B_2C_1$ is $F$, the intersection point of straight lines $B_2C_1$ and $C_2D_1$ is $G$ and the intersection point of straight lines $C_2D_1$ and $D_2A_1$ is $H$. Find the area of the square $EFGH$, knowing that the area of $ABCD$ is $1$.

Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.) [asy] unitsize(18); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle); draw((0,1)--(3,1)); draw((2,0)--(2,3)); draw((1,1)--(1,3)); label("A",(0.5,2)); label("B",(1.5,2)); label("C",(2.5,2)); label("D",(1,0.5)); [/asy]

Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.

For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that \[x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.\] Prove that $\prod_{k=1}^n x_k =1.$

Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

In a club, there are $n$ members, and they are deciding some sport training sessions, satisfying all of these requirements: [i]i)[/i] Each member attends in all training sessions; [i]ii)[/i] At each training sessions, the club are divided into $3$ groups: swimming group, cycling group, running group (each member joins exactly $1$ group and each group consists of at least $1$ person); [i]iii)[/i] For any $2$ members of the club, we can find at least $1$ session such that there are $2$ people that are not in the same group. a) Assume that $n=9$. We know that the club has ran $1$ training session and it will run $1$ more session. How many ways to divide the group for the second training session? b) Assume that $n=2022$. Find the minimum number of training sessions that the club have to run?

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

Let $a_1, a_2, \ldots, a_n$ be reals such that for all $k=1,2, \ldots, n$, $na_k \geq a_1^2+a_2^2+ \ldots+a_k^2$. Prove that there exist at least $\frac{n} {10}$ indices $k$, such that $a_k \leq 1000$.

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$. Prove, that $\angle DEB=\angle DAC$

Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$, $b_1 = 15$, and for $n \ge 1$, \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, \\ b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$. Determine the number of positive integer factors of $G$. [i]Proposed by Michael Ren[/i]

What is the number of $4$-digit numbers that contains exactly $3$ different digits that have consecutive value? Such numbers are for instance $5464$ or $2001$. Two digits in base $10$ are consecutive if their difference is $1$.

Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was $ \text{(A)}\ \text{96 cents}\qquad\text{(B)}\ \text{1.07 dollars}\qquad\text{(C)}\ \text{1.18 dollars}\qquad\text{(D)}\ \text{1.20 dollars}\qquad\text{(E)}\ \text{1.40 dollars} $

Let $ABCDEF$ be a regular hexagon with center $O$ and side length $1.$ Point $X$ is placed in the interior of the hexagon such that $\angle BXC = \angle AXE = 90^\circ.$ Compute all possible values of $OX.$

Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots. Determine the maximum number of elements in $S(a, b, c)$.

Investigate the boundary of the domain of stability ($\max \text{Re }\lambda_j < 0$) in the space of coefficients of the equation $\dddot{x} + a\ddot{x} + b\dot{x} + cx = 0$.

In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ $ and $\angle BED = \angle BDE$, then $\angle CDE = $ [asy] dot((0,0)); label("$E$",(0,0),SW); dot(dir(85)); label("$A$",dir(85),NW); dot((4,0)); label("$D$",(4,0),SE); dot((4.05677,0.648898)); label("$C$",(4.05677,0.648898),NE); draw((0,0)--dir(85)--(4.05677,0.648898)--(4,0)--cycle); dot((2,2)); label("$B$",(2,2),N); draw((0,0)--(2,2)--(4,0)); pair [] x = intersectionpoints((0,0)--(2,2)--(4,0),dir(85)--(4.05677,0.648898)); dot(x[0]); dot(x[1]); label("$F$",x[0],SE); label("$G$",x[1],SW); [/asy] $\text{(A)}\ 75^\circ \qquad \text{(B)}\ 80^\circ \qquad \text{(C)}\ 85^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 95^\circ$

For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?

Let all roots of an $n$-th degree polynomial $P(z)$ with complex coefficients lie on the unit circle in the complex plane. Prove that all roots of the polynomial $$2zP'(z)-nP(z)$$ lie on the same circle.