A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer $k \ge 0$ having the following property: for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.

Let \[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\] Compute $\Big|\lfloor 2008 X\rfloor\Big|$.

Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions: (i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$ (ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.) Find the locus of the center of circle $\Gamma$, and explain how to construct the locus.

Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.

Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?

You are given $2024$ yellow and $2024$ blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers $(a, b)$ [i]good[/i] if there exists a half-plane with exactly $a$ yellow and $b$ blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane. [i]Proposed by Anton Trygub[/i]

The infinite product $$\sqrt[3]{10}\cdot\sqrt[3]{\sqrt[3]{10}}\cdot\sqrt[3]{\sqrt[3]{\sqrt[3]{10}}}\dots$$ evaluates to a real number. What is that number? $\textbf{(A) }\sqrt{10}\qquad\textbf{(B) }\sqrt[3]{100}\qquad\textbf{(C) }\sqrt[4]{1000}\qquad\textbf{(D) }10\qquad\textbf{(E) }10\sqrt[3]{10}$

Prove that for any positive integers $p,q$ such that $\sqrt{11}>\frac{p}{q}$, the following inequality holds: \[\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.\]

A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i]. Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].

Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

There are 50 slips of paper numbered from 1 to 50. It is desired to pick 3 slips such that for any of the three numbers, divided by the greatest common divisor of the other two, the square root of the result is a rational number. How many unordered triples of slips satisfy this condition?

Find the largest integer $A$ such that, for any permutation of the natural numbers not exceeding $100$, the sum of some ten successive numbers is at least $A$.

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]

(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.

Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Let T be a finite tree graph that has more than one vertex. Let s be the largest number of vertices of a subtree $X \subset T$ for which every vertex of X has a neighbor other than X. Let t be the smallest positive integer for which each edge of T is contained in exactly t stars, and each vertex of T is contained in at most 2t - 1 stars. (That is, the stars can be represented by multiplicity.) Prove that s = t. Note: a star of T is a vertex with degree $\geq$ 3 , including its neighouring edges and vertices.

If the function $f$ satisfies the equation $f(x) + f\left ( \dfrac{1}{\sqrt[3]{1-x^3}}\right ) = x^3$ for every real $x \neq 1$, what is $f(-1)$? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ \dfrac 14 \qquad\textbf{(C)}\ \dfrac 12 \qquad\textbf{(D)}\ \dfrac 74 \qquad\textbf{(E)}\ \text{None of above} $

Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds: $\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.

Three grasshoppers $A$, $B$ and $C$ are placed on a line. Grasshopper $B$ sits at the midpoint between $A$ and $C$. Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to point $X'$, then $XY $= $YX'$). After several jumps it so happened that they returned to the three initial points (but maybe in different order). Prove that in this case $B$ returns to his initial middle position. (AK Kovaldzhy)

Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard?

If the sides of a triangle have lengths $ a, b, c$, such that $ a \plus{} b \minus{} c \equal{} 2$, and $ 2ab \minus{} c^{2} \equal{} 4$, prove that the triangle is equilateral.

Let be a natural number $ n\ge 2, n $ real numbers $ b_1,b_2,\ldots ,b_n , $ and $ n-1 $ positive real numbers $ a_1,a_2,\ldots ,a_{n-1} $ such that $ a_1+a_2+\cdots +a_{n-1} =1. $ Prove the inequality $$ b_1^2+\frac{b_2^2}{a_1} +\frac{b_3^2}{a_2} +\cdots +\frac{b_n^2}{a_{n-1}} \ge 2b_1\left( b_2+b_3+\cdots +b_n \right) , $$ and specify when equality is attained. [i]Dumitru Acu[/i]

A line $l$ intersects the segment $AB$ perpendicular to $C$. Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$. The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$. a. Prove that quadrilateral $CFDE$ is a rectangle. b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$. [asy] unitsize (2.5 cm); pair A, B, C, D, E, F, O; O = (0,0); A = (-1,0); B = (1,0); C = (-0.3,0); D = intersectionpoint(C--(C + (0,1)), Circle(O,1)); E = (C + reflect(A,D)*(C))/2; F = (C + reflect(B,D)*(C))/2; draw(Circle(O,1)); draw(Circle((A + C)/2, abs(A - C)/2)); draw(Circle((B + C)/2, abs(B - C)/2)); draw(A--B); draw(interp(C,D,-0.4)--D); draw(A--D--B); dot("$A$", A, W); dot("$B$", B, dir(0)); dot("$C$", C, SE); dot("$D$", D, NW); dot("$E$", E, SE); dot("$F$", F, SW); [/asy]

Find all real triples $(a,b,c)$ satisfying \[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]