The $14 \times 14$ chessboard squares are colored in pattern, as shown in the picture. Can you choose seven fields blacks and seven white squares of this chessboard in such a way, that there is exactly one selected field in each row and column? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8ba46030cd0f0e0511f1f9e723e5bd29e9975.png[/img]

Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that \[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\] where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.

Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10} $

Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f (999)$ ?

Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. ([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)

Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

The circle $\Omega$ centered at $O$ is the circumcircle of the triangle $ABC$. Point $D$ is chosen so that $BD \perp BC$ and points $A$ and $D$ lie in different half-planes with respect to the line $BC$. Let $E$ be a point such that $\angle ADB=\angle BDE$ and $\angle EBD+\angle ACB=90$. Point $P$ is chosen on the line $AD$ so that $OP \perp BC$. Let $Q$ be an arbitrary point on $\Omega$, and $R$ be a point on the line $BQ$ such that $PQ \parallel DR$. Prove that $\angle ARB=\angle BRE$. (All angles are oriented in the same way)

Calculate \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.

Find all polynomials $P(x)$ with integer coefficients such that $a^2+b^2-c^2$ divides $P(a)+P(b)-P(c)$, for all integers $a,b,c$.

On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Positive integers $m, n$ are given such that $\sqrt{2}<\frac{m}{n}<\sqrt{2}+\frac{1}{2}$ and $m$ is even. Prove that there exist positive integers $k<m$ and $l<n$ such that $$|\frac{k}{l}-\sqrt{2}|<\frac{m}{n}-\sqrt{2}$$

Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$. PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.

Which figure can be the images of equations $mx+ny^2=0$ and $mx^2+ny^2=1$$(m,n\neq0)$? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9kLzUzZTAwZjU1YzEyN2I3ZDJjNjcwNDQ2ZmQ5MDBmYWZlODAwNGU0LnBuZw==&rn=YWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYS5wbmc=[/img]

Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ satisfying \[ f(r)\le g(r)\quad \forall r\in \mathbb{Q} \] Does this imply $f(x)\le g(x)\quad \forall x\in \mathbb{R}$ if [list] (a)$f$ and $g$ are non-decreasing ? (b)$f$ and $g$ are continuous?[/list]

The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.

Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. The $A$-excircle, $B$-excircle, and $C$-excircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$. Show that if $M$ is the midpoint of $PA_1$, then lines $HM$ and $OI$ are parallel. [i]Michael Ren[/i]

Let $ r$, $ s$ be two positive integers and $ P$ a 'chessboard' with $ r$ rows and $ s$ columns. Let $ M$ denote the maximum value of rooks placed on $ P$ such that no two of them attack each other. (a) Determine $ M$. (b) How many ways to place $ M$ rooks on $ P$ such that no two of them attack each other? [Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]

King Tin writes the first $n$ perfect squares on the royal chalkboard, but he omits the first (so for n = $3$, he writes $4$ and $9$). His son, Prince Tin, comes along and repeats the following process until only one number remains: [i]He erases the two greatest numbers still on the board, calls them a and b, and writes the value of $\frac{ab-1}{a+b-2}$ on the board. [/i]Let $S(n)$ be the last number that Prince Tin writes on the board. Let $\lim_{n\to \infty} S(n) = r$, meaning that $r$ is the unique number such that for every $\epsilon > 0$ there exists a positive integer $N$ so that $|S(n) - r| < \epsilon$ for all $n > N$. If $r$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

Consider the polynomial $P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$ where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$ [b](a)[/b] Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$ [b](b)[/b] Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$

Find all pairs of integers $(x, y)$ such that the number \[ \frac{x^2 + y^2}{xy} \] is an integer.

Let $p$ be prime and $ k > 1$ be a divisor of $p-1$. Show that if a polynomial of degree $k$ with integer coefficients attains every possible value modulo $ p$ that is $(0,1,\dots, p-1)$ at integer inputs then its leading coefficient must be divisible by $p$. [hide=Note]Note: the leading coefficient of a polynomial of degree d is the coefficient of the $x_d$ term.[/hide]

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].