Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.

Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

Let $L_1, L_2, L_3, L_4$ be four lines in the space such that no three of them are in the same plane. Let $L_1, L_2$ intersect in $A$, $L_2,L_3$ intersect in $B$ and $L_3, L_4$ intersect in $C.$ Find minimum and maximum number of lines in the space that intersect $L_1, L_2, L_3$ and $L_4.$ Justify your answer.

Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.

In the country of Muilejistan, there exists a network of roads connecting all its cities. The network has the particular property that for any two cities, there is a unique path without backtracking (i.e., a path where the traveler never returns along the same road). The longest possible path between two cities is 600 kilometers. For instance, the path from the city of Mlar to the city of Nlar is 600 kilometers. Similarly, the path from the city of Klar to the city of Glar is also 600 kilometers. 1. If Jalim departs from Mlar towards Nlar at noon and Kalim departs from Klar towards Glar also at noon, both traveling at the same speed, prove that they meet at some point on their journey. 2. If the distance in kilometers between any two cities is an integer, prove that the distance from Glar to Mlar is even.

Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property. For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set. Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

Given a positive integer $k$ and an integer $a\equiv 3 \pmod{8}$, show that $a^m+a+2$ is divisible by $2^k$ for some positive integer $m$.

Form a square with sides of length $5$, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area [b]removed[/b] to the nearest integer.

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Let $H$ a orthocenter of $ABC$, $D$ a intersection point of $AH$ and $BC$, $E$ a intersection point of $BH$ and $AC$, and $M$ a midpoint of segment $BC$. A circle with center $E$ and radius $AE$ intersects the segment $AC$ at point $F$($\neq A$), and circumcircle of triangle $BFC$ intersects the segment $AM$ at point $S$. Let $P$($\neq D$), $Q$($\neq F$) a intersection point of circumcircle of triangle $ASD$ and $DF$, circumcircle of triangle $ASF$ and $DF$ respectively. Also, define $R$ as a intersection point of circumcircles of triangle $AHQ$ and $AEP$. Prove that $R$ lies on line $DF$.

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?

Positive integers $x, y$ satisfy the following conditions: $$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$ Show that $x = y$. Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$. [i]Proposed by Anton Trygub[/i]

Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.

The diameter of a circle of radius $R$ is divided into $2n$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true: $$f(x+y+f(y))=f(x)+2y$$ [I]proposed by F. Petrov[/i]

Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. Prove that triangles ABC', AB'C, and A'BC are similar.

Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point. EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet. Note: fresh translation

Let $a,b$ be coprime positive integers with $a$ even and $a>b$. Show that there exist infinitely many pairs $(m,n)$ of coprime positive integers such that $m\mid a^{n-1}-b^{n-1}$ and $n\mid a^{m-1}-b^{m-1}$.

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

Let $ABC$ be a triangle in the plane with $AB = 13$, $BC = 14$, $AC = 15$. Let $M_n$ denote the smallest possible value of $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim_{n \to \infty} M_n$.

Find out which of the two numbers is greater: $$\dfrac{2}{2 +\dfrac{2}{2 +\dfrac{2}{... +\dfrac{2}{2+\frac22}}}} \,\,\, \text{or} \,\,\, \dfrac{3}{3 +\dfrac{3}{3 +\dfrac{3}{... +\dfrac{3}{3+\frac33}}}}$$ (Each expression has $2022$ fraction signs.)

[u]Set 1[/u] [b]p1.[/b] Arnold is currently stationed at $(0, 0)$. He wants to buy some milk at $(3, 0)$, and also some cookies at $(0, 4)$, and then return back home at $(0, 0)$. If Arnold is very lazy and wants to minimize his walking, what is the length of the shortest path he can take? [b]p2.[/b] Dilhan selects $1$ shirt out of $3$ choices, $1$ pair of pants out of $4$ choices, and $2$ socks out of $6$ differently-colored socks. How many outfits can Dilhan select? All socks can be worn on both feet, and outfits where the only difference is that the left sock and right sock are switched are considered the same. [b]p3.[/b] What is the sum of the first $100$ odd positive integers? [b]p4.[/b] Find the sum of all the distinct prime factors of $1591$. [b]p5.[/b] Let set $S = \{1, 2, 3, 4, 5, 6\}$. From $S$, four numbers are selected, with replacement. These numbers are assembled to create a $4$-digit number. How many such $4$-digit numbers are multiples of $3$? [u]Set 2[/u] [b]p6.[/b] What is the area of a triangle with vertices at $(0, 0)$, $(7, 2)$, and $(4, 4)$? [b]p7.[/b] Call a number $n$ “warm” if $n - 1$, $n$, and $n + 1$ are all composite. Call a number $m$ “fuzzy” if $m$ may be expressed as the sum of $3$ consecutive positive integers. How many numbers less than or equal to $30$ are warm and fuzzy? [b]p8.[/b] Consider a square and hexagon of equal area. What is the square of the ratio of the side length of the hexagon to the side length of the square? [b]p9.[/b] If $x^2 + y^2 = 361$, $xy = -40$, and $x - y$ is positive, what is $x - y$? [b]p10.[/b] Each face of a cube is to be painted red, orange, yellow, green, blue, or violet, and each color must be used exactly once. Assuming rotations are indistinguishable, how many ways are there to paint the cube? [u]Set 3[/u] [b]p11.[/b] Let $D$ be the midpoint of side $BC$ of triangle $ABC$. Let $P$ be any point on segment $AD$. If $M$ is the maximum possible value of $\frac{[PAB]}{[PAC]}$ and $m$ is the minimum possible value, what is $M - m$? Note: $[PQR]$ denotes the area of triangle $PQR$. [b]p12.[/b] If the product of the positive divisors of the positive integer $n$ is $n^6$, find the sum of the $3$ smallest possible values of $n$. [b]p13.[/b] Find the product of the magnitudes of the complex roots of the equation $(x - 4)^4 +(x - 2)^4 + 14 = 0$. [b]p14.[/b] If $xy - 20x - 16y = 2016$ and $x$ and $y$ are both positive integers, what is the least possible value of $\max (x, y)$? [b]p15.[/b] A peasant is trying to escape from Chyornarus, ruled by the Tsar and his mystical faith healer. The peasant starts at $(0, 0)$ on a $6 \times 6$ unit grid, the Tsar’s palace is at $(3, 3)$, the healer is at $(2, 1)$, and the escape is at $(6, 6)$. If the peasant crosses the Tsar’s palace or the mystical faith healer, he is executed and fails to escape. The peasant’s path can only consist of moves upward and rightward along the gridlines. How many valid paths allow the peasant to escape? PS. You should use hide for answers. Rest sets have been posted [url=https://artofproblemsolving.com/community/c3h2784259p24464954]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

The incircle of $ \triangle ABC$ touches $ BC$, $ AC$, and $ AB$ at $ A_1$, $ B_1$, and $ C_1$, respectively. The line $ AA_1$ intersects the incircle at $ Q$, again. $ A_1C_1$ and $ A_1B_1$ intersect the line, passing through $ A$ and parallel to $ BC$, at $ P$ and $ R$, respectively. If $ \angle PQC_1 \equal{} 45^\circ$ and $ \angle RQB_1 \equal{} 65^\circ$, then $ \angle PQR$ will be ? $\textbf{(A)}\ 110^\circ \qquad\textbf{(B)}\ 115^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 125^\circ \qquad\textbf{(E)}\ 130^\circ$