An investor has two rectangular pieces of land of size $120\times 100$. a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house. b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.

Suppose that $p(x),q(x)$ are monic polynomials with nonnegative integer coefficients such that \[\frac{1}{5x}\ge\frac{1}{q(x)}-\frac{1}{p(x)}\ge\frac{1}{3x^2}\] for all integers $x\ge2$. Compute the minimum possible value of $p(1)\cdot q(1)$.

$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.

In a group of $n \ge 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeated and that there are $m \ge 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $\left[n+3 -\frac{18m}{n}\right]$

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

Determine all polynomials $P$ with integer coefficients satisfying \[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]

Evaluate $2023^2 -2022^2 +2021^2 -2020^2$.

A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.

a) $n$ $2\times2$ squares are drawn on the Cartesian plane. The sides of these squares are parallel to the coordinate axes. It is known that the center of any square is not an inner point of any other square. Let $\Pi$ be a rectangle such that it contains all these $n$ squares and its sides are parallel to the coordinate axes. Prove that the perimeter of $\Pi$ is greater than or equal to $4(\sqrt{n}+1)$. b) Prove the sharp estimate: the perimeter of $\Pi$ is greater than or equal to $2\lceil \sqrt{n}+1) \rceil$ (here $\lceil a\rceil$ stands for the smallest integer which is greater than or equal to $a$).

If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.

How many pairs $(a, b)$ for integers $1 \le a, b \le 2015$ which satisfy that $a$ is divisible by $b + 1$ and $2016 - a$ is divisible by $b$.

A parabola and a hyperbola are drawn on the coordinate plane. The graphs intersect at three points $A, B, C$ and the axis of the parabola is the asymptote of the hyperbola. Prove that the intersection point of the medians of the triangle $ABC$ lies on the axis of the parabola. [i]From the folklore[/i]

Find the number of real solutions $(x,y)$ to the system of equations: $$\begin{cases} \sin(x^2-y) = 0 \\ |x|+|y|=2\pi \end{cases}$$

For what integer values of $n$ is $1+n+\frac{n^2}{2}+\cdots +\frac{n^n}{n!}$ an integer?

Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$. Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$.

Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?

Let $p$ be an odd prime number and let $a_1,a_2,...,a_n \in \mathbb Q^+$ be rational numbers. Prove that \[\mathbb Q(\sqrt[p]{a_1}+\sqrt[p]{a_2}+...+\sqrt[p]{a_n})=\mathbb Q(\sqrt[p]{a_1},\sqrt[p]{a_2},...,\sqrt[p]{a_n}).\]

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find all five-digit numbers that satisfy the following conditions: 1. The number is a palindrome. 2. The middle digit is twice the value of the first digit. 3. The number is a perfect square. [i]Proposed by Tamar Turashvili, Georgia [/i]

Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $ Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $

On a circle noted $n$ points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values $n$ in which this is possible.

Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle. From each of the vertices of this triangle, we draw a tangent to the smaller circle. Prove that the length of one of these tangents equals the sum of the lengths of the two other tangents.