Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.

[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \] [i] Stoyan Atanasov[/i]

[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds: $$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$ [b]b)[/b] For $ a,b>0 $ is the above inequality still true? [i]Ilie Diaconu[/i]

Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.

Given an acute triangle $ABC$. A circle inscribed in a triangle $ABC$ with center at point $I$ touches the sides $AB, BC$ at points $C_1$ and $A_1$, respectively. The lines $A_1C_1$ and $AC$ intersect at the point $Q$. Prove that the circles circumscribed around the triangles $AIC$ and $A_1CQ$ are tangent. (Dmitry Shvetsov)

Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that $\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$

Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that \[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]

Prove that, for any integer $n\geq 2$, there exists an integer $x$ such that $3^n|x^3+2017$, but $3^{n+1}\not | x^3+2017$.

The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are [b]not[/b] collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.

A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?

Let be two real numbers $ a>0,b. $ Calculate the primitive of the function $ 0<x\mapsto\frac{bx-1}{e^{bx}+ax} . $ [i]Dan Negulescu[/i]

Write an integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors. Note: Two triangles are said to be neighbors if they have a common side. [asy]size(100); pair P=(0,0); pair Q=(2, 2*sqrt(3)); pair R=(4,0); draw(P--Q--R--cycle); pair B=midpoint(P--Q); pair A=midpoint(P--B); pair C=midpoint(B--Q); pair E=midpoint(Q--R); pair D=midpoint(Q--E); pair F=midpoint(E--R); pair H=midpoint(R--P); pair G=midpoint(R--H); pair I=midpoint(H--P); draw(A--I); draw(B--H); draw(C--G); draw(I--D); draw(H--E); draw(G--F); draw(C--D); draw(B--E); draw(A--F);[/asy]

In the regular hexagon shown below, how many diagonals are longer than the red diagonal? [asy] size(2cm); draw((0,0)--(2,0)--(3,1.732)--(2,3.464)--(0,3.464)--(-1,1.732)--cycle); draw((-1,1.732)--(2,0),red); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$ (a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.

Find solution in positive integers : $$n^5+n^4=7^m-1$$

A bag has $3$ white and $7$ black marbles. Arjun picks out one marble without replacement and then a second. What is the probability that Arjun chooses exactly $1$ white and $1$ black marble?

Pedro and Juan are playing the following game: $-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$). $-$ Each player can draw: -- $1$ rock from one of the piles, or -- $2$ rocks from one of the piles, or -- $1$ rock from each pile, or -- $2$ rock from one pile and $1$ from the other pile. Each player must perform one of these four operations in their turns. The looser is the one who takes the last rock. Pedro plays first and has a winning strategy. What are the three maximum possible values of ($X+Y$)?

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

Find the smallest possible $n$ for which there exist integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$ such that each integer between $1000$ and $2000$ (inclusive) can be written as the sum (without repetition), of one or more of the integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$.

Jason has n coins, among which at most one of them is counterfeit. The counterfeit coin (if there is any) is either heavier or lighter than a real coin. Jason’s grandfather also left him an old weighing balance, on which he can place any number of coins on either side and the balance will show which side is heavier. However, the old weighing balance is in fact really really old and can only be used 4 more times. What is the largest number $n$ for which it is possible for Jason to find the counterfeit coin (if it exist)?

Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$. [i]Author: Christopher Bradley, United Kingdom [/i]

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

17) An object is thrown directly downward from the top of a 180-meter-tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof? A) 15 m/s B) 25 m/s C) 35 m/s D) 55 m/s E) insufficient information

Find the sum of the coefficients of the polynomial $P(x) = x^4- 29x^3 + ax^2 + bx + c$, given that $P(5) = 11$, $P(11) = 17$, and $P(17) = 23$.