A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Let be a real number $ a, $ and a nondecreasing function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Prove that $ f $ is continuous in $ a $ if and only if there exists a sequence $ \left( a_n \right)_{n\ge 1} $ of real positive numbers such that $$ \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , $$ for all natural numbers $ n. $ [i]Dan Marinescu[/i]

Given $n$ different positive numbers $a_1,a_2,...,a_n$. We construct all the possible sums (from $1$ to $n$ terms). Prove that among those sums there are at least $n(n+1)/2$ different ones.

Let $P$ be a point inside a square $ABCD$ such that $PA:PB:PC$ is $1:2:3$. Determine the angle $\angle BPA$.

Let $p$ be a real number and $c\neq 0$ such that \[c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1\] for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".) Find the ordered pair $(p,c)$.

In this problem, a [i]simple polygon[/i] is a polygon that does not intersect itself and has no holes, and a [i]side[/i] of a polygon is a maximal set of collinear, consecutive line segments in the polygon. In particular, we allow two or more consecutive vertices in a simple polygon to be identical, and three or more consecutive vertices in a simple polygon to be collinear. By convention, polygons must have at least three sides. A simple polygon is [i]convex[/i] if every one of its interior angles is $180^\circ$ degrees or less. A simple polygon is concave if it is not [i]convex[/i]. Let P be the plane. Prove or disprove each of the following statements: $(a)$ There exists a function $f : P \to P$ such that for all positive integers $n \geq 4$, if $v_1, v_2, \ldots , v_n$ are the vertices of a simple concave $n$-sided polygon in some order, then $f(v_1), f(v_2), \ldots, f(v_n)$ are the vertices of a simple convex polygon in some order (which may or may not have $n$ sides). $(b)$ There exists a function $f : P \to P$ such that for all positive integers $n \geq 4$, if $v_1, v_2, \ldots , v_n$ are the vertices of a simple convex $n$-sided polygon in some order, then $f(v_1), f(v_2), \ldots, f(v_n)$ are the vertices of a simple concave polygon in some order (which may or may not have $n$ sides).

Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Consider the sequence $1, \frac12, \frac13, \frac14 ,...$ Does there exist an arithmetic progression composed of terms of this sequence (a) of length $5$, (b) of length greater than $5$ (if so, what possible length)? (G Galperin, Moscow)

Given integers $a_0,a_1, ... , a_{100}$, satisfying $a_1>a_0$, $a_1>0$, and $a_{r+2}=3 a_{r+1}-2a_r$ for $r=0, 1, ... , 98$. Prove $a_{100}>299$

The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$

Misha came to country with $n$ cities, and every $2$ cities are connected by the road. Misha want visit some cities, but he doesn`t visit one city two time. Every time, when Misha goes from city $A$ to city $B$, president of country destroy $k$ roads from city $B$(president can`t destroy road, where Misha goes). What maximal number of cities Misha can visit, no matter how president does?

Find the Green's function of the operator $d^2/dx^2-1$ and solve the equation \[\int_{-\infty}^{+\infty}e^{-|x-y|}u(y)dy=e^{-x^2}\]

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

Find all triples of positive integers \( a, b, c \) that satisfy the equation: \[ a + \frac{1}{b + \frac{1}{c}} = 20.25. \]

Lines determined by sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ intersect at point $P$. Prove that $\alpha +\gamma =\beta +\delta$ if and only if $PA\cdot PB=PC\cdot PD$, where $\alpha ,\beta ,\gamma ,\delta$ are the measures of the internal angles of vertices $A, B, C, D$ respectively.

Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.

Let $\triangle ABC$ have its vertices at $A(0, 0), B(7, 0), C(3, 4)$ in the Cartesian plane. Construct a line through the point $(6-2\sqrt 2, 3-\sqrt 2)$ that intersects segments $AC, BC$ at $P, Q$ respectively. If $[PQC] = \frac{14}3$, what is $|CP|+|CQ|$? [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 9)[/i]

Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.

Jean writes five tests and achieves the marks shown on the graph. What is her average mark on these five tests? [asy] draw(origin -- (0, 10.1)); for(int i = 0; i < 11; ++i) { draw((0, i) -- (10.5, i)); label(string(10*i), (0, i), W); } filldraw((1, 0) -- (1, 8) -- (2, 8) -- (2, 0) -- cycle, black); filldraw((3, 0) -- (3, 7) -- (4, 7) -- (4, 0) -- cycle, black); filldraw((5, 0) -- (5, 6) -- (6, 6) -- (6, 0) -- cycle, black); filldraw((7, 0) -- (7, 9) -- (8, 9) -- (8, 0) -- cycle, black); filldraw((9, 0) -- (9, 8) -- (10, 8) -- (10, 0) -- cycle, black); label("Test Marks", (5, 0), S); label(rotate(90)*"Marks out of 100", (-2, 5), W); [/asy] $\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 79$

Let $x, y$, and $z$ be three not necessarily real numbers that satisfy the following system of equations: $x^3 -4 = (2y +1)^2$ $y^3 -4 = (2z +1)^2$ $z^3 -4 = (2x +1)^2$. Find the greatest possible real value of $(x -1)(y -1)(z -1)$.

There are $n$ dots on the plane such that no three dots are collinear. Each dot is assigned a $0$ or a $1$. Each pair of dots is connected by a line segment. If the endpoints of a line segment are two dots with the same number, then the segment is assigned a $0$. Otherwise, the segment is assigned a $1$. Find all $n$ such that it is possible to assign $0$'s and $1$'s to the $n$ dots in a way that the corresponding line segments are assigned equally many $0$'s as $1$'s.

We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is [i]excellent[/i] if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$, at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$.) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.