Assume we can fill a table $n\times n$ with all numbers $1,2,\ldots,n^2-1,n^2$ in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers $n$.

$300$ people took part in the drawing for the main prize of the television lottery. They lined up in a circle, then, starting with someone who received number $1$, they began to count them. Moreover, every third person dropped out every time. (So, in the first round, everyone with numbers divisible by $3$ dropped out). The counting continued until there was only one person left. (It is clear that more than one circle was made). This person received the main prize. (It “accidentally” turned out to be the TV director’s mother-in-law). What number did this person have in the initial lineup?

The real positive numbers $a_1, a_2,a_3$ are three consecutive terms of an arithmetic progression, and similarly, $b_1, b_2, b_3$ are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths $a_1, a_2, a_3$ as bases, and other three segments of lengths $b_1, b_2, b_3$ as altitudes, to construct three rectangles of equal area ?

In the table shown in the figure, Zosia replaced eight numbers with their negatives. It turned out that each row and each column contained exactly two negative numbers. Prove that after this change, the sum of all sixteen numbers in the table is equal to $0$. [center] [img] https://wiki-images.artofproblemsolving.com//2/2e/17-3-5.png [/img] [/center]

We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)

Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.

What is the positive difference between the largest possible two-digit integer and the smallest possible three-digit integer? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad\textbf{(E) }9$

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer

There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$. Find $100m+n$ [i]Proposed by Yannick Yao[/i]

The circle $\omega$ with center $O$ has given. From an arbitrary point $T$ outside of $\omega$ draw tangents $TB$ and $TC$ to it. $K$ and $H$ are on $TB$ and $TC$ respectively. [b]a)[/b] $B'$ and $C'$ are the second intersection point of $OB$ and $OC$ with $\omega$ respectively. $K'$ and $H'$ are on angle bisectors of $\angle BCO$ and $\angle CBO$ respectively such that $KK' \bot BC$ and $HH'\bot BC$. Prove that $K,H',B'$ are collinear if and only if $H,K',C'$ are collinear. [b]b)[/b] Consider there exist two circle in $TBC$ such that they are tangent two each other at $J$ and both of them are tangent to $\omega$.and one of them is tangent to $TB$ at $K$ and other one is tangent to $TC$ at $H$. Prove that two quadrilateral $BKJI$ and $CHJI$ are cyclic ($I$ is incenter of triangle $OBC$).

The teacher writes a $2002$-digit number consisting only of digits $9$ on the blackboard. The first student factors this number as $ab$ with $a > 1$ and $b > 1$ and replaces it on the blackboard by two numbers $a'$ and $b'$ with $|a-a'| = |b-b'| = 2$. The second student chooses one of the numbers on the blackboard, factors it as $cd$ with $c > 1$ and $d > 1$ and replaces the chosen number by two numbers $c'$ and $d'$ with $|c-c'| = |d-d'| = 2$, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to $9$?

Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal. Determine the smallest possible degree of $f$. (Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.) [i]Ankan Bhattacharya[/i]

There is a school with $n$ students. Suppose that every student has exactly $2023$ friends and every couple of student that are not friends has exactly $2022$ friends in common. Then find all values of $n$

Each side of a convex $2019$-gon polygon is dyed with red, yellow and blue, and there are exactly $673$ sides of each kind of color. Prove that there exists at least one way to draw $2016$ diagonals to divide the convex $2019$-gon polygon into $2017$ triangles, such that any two of the $2016$ diagonals don't have intersection inside the $2019$-gon polygon,and for any triangle in all the $2017$ triangles, the colors of the three sides of the triangle are all the same, either totally different.

At the midpoint of line segment $ AB$ which is $ p$ units long, a perpendicular $ MR$ is erected with length $ q$ units. An arc is described from $ R$ with a radius equal to $ \frac{1}{2}AB$, meeting $ AB$ at $ T$. Then $ AT$ and $ TB$ are the roots of: $ \textbf{(A)}\ x^2\plus{}px\plus{}q^2\equal{}0 \\ \textbf{(B)}\ x^2\minus{}px\plus{}q^2\equal{}0 \\ \textbf{(C)}\ x^2\plus{}px\minus{}q^2\equal{}0 \\ \textbf{(D)}\ x^2\minus{}px\minus{}q^2\equal{}0 \\ \textbf{(E)}\ x^2\minus{}px\plus{}q\equal{}0$

For each real number $a$ and integer $n \ge 1$ prove the inequality $a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)$ and find the cases of equality.

How many triangles formed by three vertices of a regular $17$-gon are obtuse? $\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$

Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

Find all polynomials $f$ such that $f$ has non-negative integer coefficients, $f(1)=7$ and $f(2)=2017$.

Show that there are only finitely many integral solutions to \[ 3^m - 1 = 2^n \] and find them.

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that: [LIST] [*] in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and [/*] [*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.[/*] [/LIST] [b]Note.[/b] The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

Let $(\Omega, \mathcal{A},\mathbb{P})$ be a standard probability space, and $\mathcal{X}$ be the set of all bounded random variables. For $t>0$, defined the mapping $R_t$ by \[R_t(X)=t\log \mathbb{E}[\exp(X/t)], \quad X \in \mathcal{X},\] and for $t \in (0,1)$ define the mapping $Q_t$ by \[Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\}, \quad X \in \mathcal{X}.\] For two mappings $f,g:\mathcal{X} \to [-\infty,\infty)$, defined the operator $\square$ by \[f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\}, \quad X \in \mathcal{X}.\] (a) Show that, for $t,s>0$, \[R_t \square R_s=R_{t+s}.\] (b) Show that, for $t,s>0$ with $t+s<1$, \[Q_t \square Q_s=Q_{t+s}.\]