Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $p$ be an odd prime and let $$F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.$$Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\ldots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$.

Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker. If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer. [b]Comment.[/b] Batman and Robin decide their common strategy at the beginning. [img]https://i.imgur.com/PeLBQNt.png[/img]

Investigate whether a chess knight can traverse a $4 \times 4$ mini-chessboard so that it reaches each of the $16$ squares only once. Note: the drawing below shows the endpoints of the eight possible moves of the knight $(C)$ on a chessboard of size $8 \times 8$. [asy] unitsize(0.4 cm); int i; fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((2,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("C", (3.5,4.5), fontsize(8)); [/asy]

There are $3$ sticks of each color between blue, red and green, such that we can make a triangle $T$ with sides sticks with all different colors. Dana makes $2$ two arrangements, she starts with $T$ and uses the other six sticks to extend the sides of $T$, as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same. [asy]size(300); pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K; A = (0, 0); B = (1, 0); C=(-0.5,2); D=(-1.1063,4.4254); M=(-1.7369,3.6492); N=(3.5,0); P=(-2.0616,0); Q=(0.2425,-0.9701); R=(1.6,-0.8); S=(7.5164,0.8552); T=(8.5064,0.8552); U=(7.0214,2.8352); V=(8.1167,-1.546); W=(9.731,-0.7776); X=(10.5474,0.8552); Y=(6.7813,3.7956); Z=(6.4274,3.6272); K=(5.0414,0.8552); draw(A--B, blue); label("$b$", (A + B) / 2, dir(270), fontsize(10)); label("$g$", (B+C) / 2, dir(10), fontsize(10)); label("$r$", (A+C) / 2, dir(230), fontsize(10)); draw(B--C,green); draw(D--C,green); label("$g$", (C + D) / 2, dir(10), fontsize(10)); draw(C--A,red); label("$r$", (C + M) / 2, dir(200), fontsize(10)); draw(B--N,green); label("$g$", (B + N) / 2, dir(70), fontsize(10)); draw(A--P,red); label("$r$", (A+P) / 2, dir(70), fontsize(10)); draw(A--Q,blue); label("$b$", (A+Q) / 2, dir(540), fontsize(10)); draw(B--R,blue); draw(C--M,red); label("$b$", (B+R) / 2, dir(600), fontsize(10)); draw(Q--R--N--D--M--P--Q, dashed); draw(Y--Z--K--V--W--X--Y, dashed); draw(S--T,blue); draw(U--T,green); draw(U--S,red); draw(T--W,red); draw(T--X,red); draw(S--K,green); draw(S--V,green); draw(Y--U,blue); draw(U--Z,blue); label("$b$", (Y+U) / 2, dir(0), fontsize(10)); label("$b$", (U+Z) / 2, dir(200), fontsize(10)); label("$b$", (S+T) / 2, dir(100), fontsize(10)); label("$r$", (S+U) / 2, dir(200), fontsize(10)); label("$r$", (T+W) / 2, dir(70), fontsize(10)); label("$r$", (T+X) / 2, dir(70), fontsize(10)); label("$g$", (U+T) / 2, dir(70), fontsize(10)); label("$g$", (S+K) / 2, dir(70), fontsize(10)); label("$g$", (V+S) / 2, dir(30), fontsize(10)); [/asy]

A bagel is a loop of $2a+2b+4$ unit squares which can be obtained by cutting a concentric $a\times b$ hole out of an $(a +2)\times (b+2)$ rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length $a+2$ of the rectangle.) Consider an infinite grid of unit square cells. For each even integer $n \ge 8$, a bakery of order $n$ is a finite set of cells $ S$ such that, for every $n$-cell bagel $B$ in the grid, there exists a congruent copy of $B$ all of whose cells are in $S$. (The copy can be translated and rotated.) We denote by $f(n)$ the smallest possible number of cells in a bakery of order $ n$. Find a real number $\alpha$ such that, for all sufficiently large even integers $n \ge 8$, we have $$\frac{1}{100}<\frac{f (n)}{n^ {\alpha}}<100$$ [i]Proposed by Nikolai Beluhov[/i]

Let $ABC$ be a triangle with circumcenter $O$. Points $P$ and $Q$ are interior to sides $CA$ and $AB$, respectively. Circle $\omega$ passes through the midpoints of segments $BP$, $CQ$, $PQ$. Prove that if line $PQ$ is tangent to circle $\omega$, then $OP = OQ$.

If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $ A$ and $ B$, respectively. Each polygon had a side length of $ 2$. Which of the following is true? $ \textbf{(A)}\ A\equal{}\frac{25}{49}B\qquad \textbf{(B)}\ A\equal{}\frac{5}{7}B\qquad \textbf{(C)}\ A\equal{}B\qquad \textbf{(D)}\ A\equal{}\frac{7}{5}B\qquad \textbf{(E)}\ A\equal{}\frac{49}{25}B$

$a_1,…,a_n$ are real numbers such that $a_1+…+a_n=0$ and $|a_1|+…+|a_n|=1$. Prove that : $$|a_1+2a_2+…+na_n| \leq \frac{n-1}{2}$$

In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.

Find $n$ so that $4^{4^{4^2}}=2^{8^n}$.

(A.Khachaturyan, 10--11) a) All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid? b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?

Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$-altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$. Show that the circumcircles of the two triangles $ACM$, $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$.

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$. by Iman Maghsoudi