$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

Show that for any positive integer $n$, \[\frac{\sigma(n!)}{n!}\ge \sum_{k=1}^{n}\frac{1}{k}.\]

Let $ E$ be a system of $ n^2 \plus{} 1$ closed intervals of the real line. Show that $ E$ has either a subsystem consisting of $ n \plus{} 1$ elements which are monotonically ordered with respect to inclusion or a subsystem consisting of $ n \plus{} 1$ elements none of which contains another element of the subsystem.

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

Let $P$ be a point inside a triangle $ABC$. Lines $AP,BP,CP$ meet the opposite sides of the triangle at points $A',B',C'$ respectively. Denote $x =\frac{AP}{PA'}, y = \frac{BP}{PB'}$ and $z = \frac{CP}{PC'}$. Prove that $xyz = x+y+z+2$.

Given that $x^2+5x+6=20$, find the value of $3x^2+15x+17$.

Let $x_0$, $x_1$, $x_2$, and $x_3$ be complex numbers forming a square centered at $0$ in the complex plane with side length $2$. For each $0\leq k\leq 3$, there are four more complex numbers $z_{4k}, z_{4k+1}$, $z_{4k+2}$, and $z_{4k+3}$ forming a square centered at $x_k$ with side length $\sqrt 2$. Given that $\prod_{i=0}^{15} z_i$ is a positive integer, how many possible values could it take? [i]Proposed by Hari Desikan[/i]

The sum of distinct real roots of the polynomial $x^5+x^4-4x^3-7x^2-7x-2$ is $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -2 \qquad\textbf{(E)}\ 7$

In a checkered square of size $2021\times 2021$ all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast as possible. How many steps will this take? [i]Ivan Yashchenko[/i]

Find all functions $f : R \to R$ such that $$f(x + zf(y)) = f(x) + zf(y), $$ for all $x, y, z \in R$.

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.

For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality $$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$

A convex polygon $\mathcal{P}$ with a center of symmetry $O{}$ is drawn in the plane. Prove that it is possible to place a rhombus in $\mathcal{P}$ whose image following a homothety of factor two centered at $O$ contains $\mathcal{P}$. [i]Proposed by I. Bogdanov, S. Gerdzhikov and N. Nikolov[/i]

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.

For each pair of integers \( j, k \geq 2 \), define the function \( f_{jk} : \mathbb{R} \to \mathbb{R} \) given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers \( j, k \geq 2 \), there exists a unique real number \( p_{jk} \in (0, 1) \) such that \( f_{jk}(p_{jk}) = p_{jk} \). Furthermore, defining \( \lambda_{jk} := f'_{jk}(p_{jk}) \), prove that \( \lambda_{jk} > 1 \). (b) Prove that \( p^j_{jk} = 1 - p_{kj} \) for any integers \( j, k \geq 2 \). (c) Prove that \( \lambda_{jk} = \lambda_{kj} \) for any integers \( j, k \geq 2 \).

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in: $ \textbf{(A)}\ \text{none of the quadrants} \qquad\textbf{(B)}\ \text{quadrant I only} \qquad\textbf{(C)}\ \text{quadrants I, II only} \qquad$ $\textbf{(D)}\ \text{quadrants I, II, III only} \qquad\textbf{(E)}\ \text{each of the quadrants} $

Let $M$ be the point of intersection of the diagonals, and $K$ be the point of intersection of the bisectors of the angles $B$ and $C$ of the convex quadrilateral $ABCD$. Prove that points $A, B, M, K$ lie on the same circle if the following relation holds: $|AB|=|BC|=|CD|$

The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio $2:1$ (closer to the feet) all lie on a sphere.

The function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called [i]“Sozopolian”[/i], if it satisfies the following two properties: For each two $x,y\in \mathbb{Z}$ which aren’t multiples of 17 the number $f(xy)-f(x)-f(y)$ is divisible by 8; For $\forall x\in \mathbb{Z}$ the number $f(x+17)-f(x)$ is divisible by 8. Does there exist a [i]Sozopolian[/i] function for which a) $f(2)=1; \quad$ b) $f(3)=1$?

The figure below shows a hexagonal net formed by many congruent equilateral triangles. Taking turns, Gabriel and Arnaldo play a game as follows. On his turn, the player colors in a segment, including the endpoints, following these three rules: i) The endpoints must coincide with vertices of the marked equilateral triangles. ii) The segment must be made up of one or more of the sides of the triangles. iii) The segment cannot contain any point (endpoints included) of a previously colored segment. Gabriel plays first, and the player that cannot make a legal move loses. Find a winning strategy and describe it.

Let $n \ne 0$ be a natural number. A sequence of numbers is briefly called a sequence “$F_n$” if $n$ different numbers $z_1$, $z_2$, $...$, $z_n$ exist so that the following conditions are fulfilled: (1) Each term of the sequence is one of the numbers $z_1$, $z_2$, $...$, $z_n$. (2) Each of the numbers $z_1$, $z_2$, $...$, $z_n$ occurs at least once in the sequence. (3) Any two immediately consecutive members of the sequence are different numbers. (4) No subsequence of the sequence has the form $\{a, b, a, b\}$ with $a \ne b$. Note: A subsequence of a given sequence $\{x_1, x_2, x_3, ...\}$ or $\{x_1, x_2, x_3, ..., x_s\}$ is called any sequence of the form $\{x_{m1}, x_{m2}, x_{m3}, ...\}$ or $\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\}$ with natural numbers $m_1 < m_2 < m_3 < ...$ Answer the following questions: a) Given $n$, are there sequences $F_n$ of arbitrarily long length? b) If question (a) is answered in the negative for an $n$: What is the largest possible number of terms that a sequence $F_n$ can have (given $n$)?

Let $ABC$ be a triangle with $AB < BC$. The bisector of angle $C$ meets the line parallel to $AC$ and passing through $B$, at point $P$. The tangent at $B$ to the circumcircle of $ABC$ meets this bisector at point $R$. Let $R'$ be the reflection of $R$ with respect to $AB$. Prove that $\angle R'P B = \angle RPA$.

Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form \[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\] with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.