What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?

For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$. How many different residues modulo $1000$ do you get from $n!!$ for $n= 1, 3, 5, …$?

Let $ f(x) = x^3 + ax + b $, with $ a \ne b $, and suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. Find $f(1)$.

Prove that for arbitary positive integer $ n\geq 4$, there exists a permutation of the subsets that contain at least two elements of the set $ G_{n} \equal{} \{1,2,3,\cdots,n\}$: $ P_{1},P_{2},\cdots,P_{2^n \minus{} n \minus{} 1}$ such that $ |P_{i}\cap P_{i \plus{} 1}| \equal{} 2,i \equal{} 1,2,\cdots,2^n \minus{} n \minus{} 2.$

Find the shortest distance between the points $(3,5)$ and $(7,8)$.

Find the number of ways to color $n \times m$ board with white and black colors such that any $2 \times 2$ square contains the same number of black and white cells.

$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

Let $S$ be the set of $2\times2$-matrices over $\mathbb{F}_{p}$ with trace $1$ and determinant $0$. Determine $|S|$.

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$.

Given any six points in the plane, no three collinear, show that we can always find three which form an obtuse-angled triangle with one angle at least $120^o$.

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles. (Andrey Mostovy)

Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions: (1) $f(0) = 0, f(1) = 1,$ (2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$ Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$

Let $M$ be inside segment $BC$ in triangle $\triangle ABC$. $(ABM)$ cuts $AC$ in $A$ and $N$. Construct the circle through $A,N$ and tangent to $BC$ in $P$. Prove that $\measuredangle BAP=\measuredangle PNM$.

Is it possible to arrange all the integers from $0$ to $9$ in circles so that the sum of three numbers along any of the six segments is the same? [img]https://cdn.artofproblemsolving.com/attachments/c/1/1a577fb4a607c395f5cc07b63653307b569b95.png[/img]

We say that a number $n \ge 2$ has the property $(P)$ if, in its prime factorization, at least one of the factors has an exponent $3$. a) Determine the smallest number $N$ with the property that, no matter how we choose $N$ consecutive natural numbers, at least one of them has the property $(P).$ b) Determine the smallest $15$ consecutive numbers $a_1, a_2, \ldots, a_{15}$ that do not have the property $(P),$ such that the sum of the numbers $5 a_1, 5 a_2, \ldots, 5 a_{15}$ is a number with the property $(P).$

A teacher plays the game “Duck-Goose-Goose” with his class. The game is played as follows: All the students stand in a circle and the teacher walks around the circle. As he passes each student, he taps the student on the head and declares her a ‘duck’ or a ‘goose’. Any student named a ‘goose’ leaves the circle immediately. Starting with the first student, the teacher tags students in the pattern: duck, goose, goose, duck, goose, goose, etc., and continues around the circle (re-tagging some former ducks as geese) until only one student remains. This remaining student is the winner. For instance, if there are 8 students, the game proceeds as follows: student 1 (duck), student 2 (goose), student 3 (goose), student 4 (duck), student 5 (goose), student 6 (goose), student 7 (duck), student 8 (goose), student 1 (goose), student 4 (duck), student 7 (goose) and student 4 is the winner. Find, with proof, all values of $n$ with $n>2$ such that if the circle starts with $n$ students, then the $n$th student is the winner.

Let $a,b,c,d$ be non-negative integers. $(1)$ If $a^2+b^2-cd^2=2022 ,$ find the minimum of $a+b+c+d;$ $(1)$ If $a^2-b^2+cd^2=2022 ,$ find the minimum of $a+b+c+d .$

Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square.

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $ A$ and $ B$, respectively. Each polygon had a side length of $ 2$. Which of the following is true? $ \textbf{(A)}\ A\equal{}\frac{25}{49}B\qquad \textbf{(B)}\ A\equal{}\frac{5}{7}B\qquad \textbf{(C)}\ A\equal{}B\qquad \textbf{(D)}\ A\equal{}\frac{7}{5}B\qquad \textbf{(E)}\ A\equal{}\frac{49}{25}B$

we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down) for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$. for example we have such table for $n=5$ 1 0 3 3 4 1 3 2 1 1 0 1 0 1 0 2 1 0 0 0 1 0 0 0 0 prove that for $n>5$ there is no $selfish$ table

Does every monotone increasing function $f : \mathbb[0,1] \rightarrow \mathbb[0,1]$ have a fixed point? What about every monotone decreasing function?

let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

Let $A_1A_2A_3 \cdots A_{13}$ be a regular $13$-gon, and let lines $A_6A_7$ and $A_8A_9$ intersect at $B$. Show that the shaded area below is half the area of the entire polygon (including triangle $A_7A_8B$) [asy] size(2inch); pair get_point(int ind) { return dir(90 + (ind + 12) * 360 / 13); } void fill_pts(int[] points) { path p = get_point(points[0]); for (int i = 1; i < points.length; ++i) { p = p -- get_point(points[i]); } p = p -- cycle; filldraw(p, RGB(160, 160, 160), black); } fill_pts(new int[]{12, 13, 2, 3}); fill_pts(new int[]{10, 11, 4, 5}); fill_pts(new int[]{8, 9, 6, 7}); draw(polygon(13)); for (int i = 1; i <= 13; ++i) { label(get_point(i), "$A_{" + (string)(i) + "}$", get_point(i)); dot(get_point(i)); } pair B = intersectionpoint(get_point(8) -- 3 * (get_point(8) - get_point(9)) + get_point(9), get_point(7) -- 3 * (get_point(7) - get_point(6)) + get_point(6)); draw(get_point(7) -- B -- get_point(8)); label("$B$", B, S); dot(B); [/asy]