For which values of $n$ is it possible to construct an $n$ by $n$ by $n$ cube with $n^3$ unit cubes, each of which is black or white, such that each cube shares a common face with exactly three cubes of the opposite colour? (S Tokarev)

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

Let the incircle of triangle $ABC$ touches the sides $AB,BC,CA$ in $C_1,A_1,B_1$ respectively. If $A_1C_1$ cuts the parallel to $BC$ from $A$ at $K$ prove that $\angle KB_1A_1=90.$

Nine points are given on a plane, no three of which lie on a line. Any two of these points are joined by a segment. Is it possible to color these segments by several colors in such a way that, for each color, there are exactly three segments of that color and these three segments form a triangle?

An $m\times n(m,n\in \mathbb{Z}_+)$ rectangle is divided into some smaller squares. All sides of each square are parallel to the sides of the rectangle, and the length of each side is an integer. Determine the minimum value of the sum of the lengths of sides of these squares.

Triangle $ABC$ has sidelengths $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with sidelengths $AG$, $BG$, and $CG$

If for function $f$ holds that $$f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0$$ for any $\in\mathbb{R}$, prove that $f$ is periodic and find one period of her.

19. Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ ($1$, powers of $2$, and powers of $2017$ are thus contained in $S$). Compute $\sum_{s\in S}\frac1s$. 20. Let $\mathcal{V}$ be the volume enclosed by the graph $$x^ {2016} + y^{2016} + z^2 = 2016$$ Find $\mathcal{V}$ rounded to the nearest multiple of ten. 21. Zlatan has $2017$ socks of various colours. He wants to proudly display one sock of each of the colours, and he counts that there are $N$ ways to select socks from his collection for display. Given this information, what is the maximum value of $N$?

[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous. [b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.

Let $AB$ and $CD$ be two perpendicular diameters of a circle with centre $O$. Consider a point $M$ on the diameter $AB$, different from $A$ and $B$. The line $CM$ cuts the circle again at $N$. The tangent at $N$ to the circle and the perpendicular at $M$ to $AM$ intersect at $P$. Show that $OP = CM$.

Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds $$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$ where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$. [i](Serbia)[/i]

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

Fill in the grid with the numbers 1 to 6 so that each number appears exactly once in each row and column. A horizontal gray line marks any cell when it is the middle cell of the three consecutive cells with the largest sum in that row. Similarly, a vertical gray line marks any cell when it is the middle of the three consecutive cells with the largest sum in that column. If there is a tie, multiple lines are drawn in the row or column. A cell can have both lines drawn, with the appearance of a plus sign. [asy] // Change this to see the solution bool DRAW_SOLUTION = true; int n = 6; real LINE_WIDTH = 0.3; void drawHLine(int x, int y) { fill((x,y+0.5-LINE_WIDTH/2)--(x,y+0.5+LINE_WIDTH/2)--(x+1,y+0.5+LINE_WIDTH/2)--(x+1,y+0.5-LINE_WIDTH/2)--cycle, gray(0.8)); } void drawVLine(int x, int y) { fill((x+0.5-LINE_WIDTH/2,y)--(x+0.5+LINE_WIDTH/2,y)--(x+0.5+LINE_WIDTH/2,y+1)--(x+0.5-LINE_WIDTH/2,y+1)--cycle, gray(0.8)); } void drawNum(int x, int y, int num) { label(scale(1.5)*string(num), (x+0.5,y+0.5)); } void drawSolNum(int x, int y, int num) { if (DRAW_SOLUTION) { drawNum(x, y, num); } } drawHLine(2,0); drawHLine(4,1); drawHLine(1,2); drawHLine(3,2); drawHLine(4,3); drawHLine(2,4); drawHLine(3,5); drawVLine(0,4); drawVLine(1,3); drawVLine(2,1); drawVLine(2,3); drawVLine(3,4); drawVLine(4,1); drawVLine(5,2); drawNum(0, 0, 5); drawNum(4, 0, 3); drawNum(1, 2, 2); drawNum(3, 3, 4); for(int i = 0; i <= 6; i += 1) { draw((i,0)--(i,6)); draw((0,i)--(6,i)); } [/asy]

In the figure, $AB \perp BC$, $BC \perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$. In which one of the following cases is the area of $ABCD$ an integer? [asy] size(170); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2)); draw(unitcircle); dot(O); draw(A--B--C--D--A); label("$A$",A,dir(A)); label("$B$",B,dir(B)); label("$C$",C,dir(C)); label("$D$",D,dir(D)); label("$O$",O,N); [/asy] $ \textbf{(A)}\ AB=3, CD=1\qquad\textbf{(B)}\ AB=5, CD=2\qquad\textbf{(C)}\ AB=7, CD=3\qquad\textbf{(D)}\ AB=9, CD=4\qquad\textbf{(E)}\ AB=11, CD=5 $

Find the exact value of $\sin 36^o$.

Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$. [i]Proposed by Anthony Yang[/i]

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Two consecutive positive integers $n$ and $n+1$ have the property that they both have $6$ divisors but a different number of distinct prime factors. Find the sum of the possible values of $n$. [i]Proposed by Harry Kim[/i]

Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, $BC = 6$. The angle bisector of $C$ intersects side $AB$ at $X$. Points $M$ and $N$ are drawn on sides $BC$ and $AC$, respectively, such that $\overline{XM} \parallel \overline{AC}$ and $\overline{XN} \parallel \overline{BC}$. Compute the length $MN$.

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways: 1) take a coin from an arbitrary pile; 2) select a pile and divide it into two non-empty piles. The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.

An L-shaped figure composed of $4$ unit squares (such as shown in the picture) we call L-dominoes. [img]https://cdn.artofproblemsolving.com/attachments/b/2/064b7c7de496f981cd937cbb7392efc1066420.png[/img] Determine the maximum number of L-dominoes that can be placed on a board of dimensions $n \times n$, where $n$ is natural number, so that no two dominoes overlap and it is possible get from the upper left to the lower right corner of the board by moving only across those squares that are not covered by dominoes. (By moving, we move from someone of the square on it the neighboring square, i.e. the square with which it shares the page). Note: L-Dominoes can be rotated as well as flipped, giving an symmetrical figure wrt axis compared to the one shown in the picture.

Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue. [list=1] [*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there? [*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color. [*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red? [/list]

Let \(a_1, a_2, \ldots, a_n\) be integers such that \(a_1 > a_2 > \cdots > a_n > 1\). Let \(M = \operatorname{lcm} \left( a_1, a_2, \ldots, a_n \right)\). For any finite nonempty set $X$ of positive integers, define \[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \] Such a set $X$ is called [i]minimal[/i] if for every proper subset $Y$ of it, $f(Y) < f(X)$ always holds. Suppose $X$ is minimal and $f(X) \geqslant \frac{2}{a_n}$. Prove that \[ |X| \leqslant f(X) \cdot M. \]