Find all positive integers $ n$ such that $ n^4\minus{}4n^3\plus{}22n^2\minus{}36n\plus{}18$ is a perfect square.

Let $x,y,z$ be positive reals and $x \geq y \geq z$. Prove that \[\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2\]

In $ n$-dimensional Euclidean space, the square of the two-dimensional Lebesgue measure of a bounded, closed, (two-dimensional) planar set is equal to the sum of the squares of the measures of the orthogonal projections of the given set on the $ n$-coordinate hyperplanes. [i]L. Tamassy[/i]

There are $2011$ stones, whose weights are positive integers. If it is possible to divide these stones into $n$ groups not containing two stones with one weighs two times of the other, what is the least possible value of $n$? $\textbf{(A)}\ 102 \qquad\textbf{(B)}\ 51 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ \text{None}$

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

Let $ a,b,c,d$ be four distinct positive integers in arithmetic progression. Prove that $ abcd$ is not a perfect square.

An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?

[b]p16[/b] Madelyn is being paid $\$50$/hour to find useful [i]Non-Functional Trios[/i], where a Non-Functional Trio is defined as an ordered triple of distinct real numbers $(a, b, c)$, and a Non- Functional Trio is [i]useful [/i] if $(a, b)$, $(b, c)$, and $(c, a)$ are collinear in the Cartesian plane. Currently, she’s working on the case $a+b+c = 2022$. Find the number of useful Non-Functional Trios $(a, b, c)$ such that $a + b + c = 2022$. [b]p17[/b] Let $p(x) = x^2 - k$, where $k$ is an integer strictly less than $250$. Find the largest possible value of k such that there exist distinct integers $a, b$ with $p(a) = b$ and $p(b) = a$. [b]p18[/b] Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$ such that $AB = 13$, $BC = 14$, and $CA = 15$. $BH$ and $CH$ meet $\Gamma$ again at points $D$ and $E$, respectively, and $DE$ meets $AB$ and $AC$ at $F$ and $G$, respectively. The circumcircles of triangles $ABG$ and $ACF$ meet BC again at points $P$ and $Q$. If $PQ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$, find $a + b$.

Define a [i]rooted tree[/i] to be a tree $T$ with a singular node designated as the [i]root[/i] of $T$. (Note that every node in the tree can have an arbitrary number of children.) Each vertex adjacent to the root node of $T$ is itself the root of some tree called a [i]maximal subtree[/i] of $T$. Say two rooted trees $T_1$ and $T_2$ are [i]similar[/i] if there exists some way to cycle the maximal subtrees of $T_1$ to get $T_2$. For example, the first pair of trees below are similar but the second pair are not. How many rooted trees with $2019$ nodes are there up to similarity? [center] [img=500x100]https://i.imgur.com/8axcDvz.png[/img] [/center]

Let there be a regular hexagon with sidelength $1$. Find the greatest integer $n\geq2$ for which there exist $n{}$ points inside or on the sides of the hexagon such that the distance between every two points is no less than $\sqrt{2}$.

Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.

Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$. [i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]

Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.

let $F_q$ be a finite field with char ≠ 2, and let $V = F_q \times F_q$ be the 2-dimensional vector space over $F_q$. Let L ⊂ V be a subset containing lines in all directions. The order of a point in V is the number of lines in L that pass through the point. Prove that L contains at least q lines having a third-order point.

Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.

Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

$ \lim_{n\to\infty} n2^n\int_1^n \frac{dx}{\left( 1+x^2\right)^n} $ [i][i]Bogdan Enescu[/i][/i]

[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$. [b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic. [b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane? [b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$. [b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$. [b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].