The tangent lines to the circumcircle of triangle ABC passing through vertices $B$ and $C$ intersect at point $F$. Points $M$, $L$ and $N$ are the feet of the perpendiculars from vertex $A$ to the lines $FB$, $FC$ and $BC$ respectively. Show that $AM+AL \geq 2AN$

Let $K, L, M$ denote three points on the sides $BC$, $AB$ and $BC$ of $\triangle{ABC}$, so that $ALKM$ is a parallelogram. Points $S$ and $T$ are chosen on lines $KL$ and $KM$ respectively, so that the quadrilaterals $AKBS$ and $AKCT$ are both cyclic. Prove that $MLST$ is cyclic if and only if $K$ is the midpoint of $BC$.

Given $n\geq2$, let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$, it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$. Prove that $|\mathcal{A}| \leq 2^{n-2}.$

Jerry has a four-sided die, a six-sided die, and an eight-sided die. Each die is numbered starting at one. Jerry rolls the three dice simultaneously. What is the probability that they all show different numbers? $\mathrm{(A) \ } \frac{35}{48} \qquad \mathrm{(B) \ } \frac{35}{64} \qquad \mathrm {(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{5}{12} \qquad \mathrm{(E) \ } \frac{5}{8}$

In a triangle \( ABC \), \( D \) is the foot of the altitude from \( C \). Let \( P \in \overline{CD} \). \( Q \) is the intersection of \( \overline{AP} \) and \( \overline{CB} \), and \( R \) is the intersection of \( \overline{BP} \) and \( \overline{CA} \). Prove that \( \angle RDC = \angle QDC \).

Given $a_0, a_1, ... , a_n$. It is known that $$a_0=a_n=0, a_{k-1}-2a_k+a_{k+1}\ge 0$$ for all $k = 1, 2, ... , k-1$.Prove that all the numbers are nonnegative.

Let $x_1, x_2, ... , x_n$ be a real numbers such that\\ 1) $1 \le x_1, x_2, ... , x_n \le 160$ 2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$ Find the largest possible $n$.

There are $64$ towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is no algorithm which enables us to do so in less than $2016$ questions. (Proposed by Konstantin Knop)

Let $\Omega$ be a circle, and $B, C$ are two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a fixed circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.

Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the foot of the angle bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$. (A.Akopjan)

Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

Find the greatest integer $n$ such that $10^n$ divides $$\frac{2^{10^5} 5^{2^{10}}}{10^{5^2}}$$

[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class $\mathcal{C}$ of the partition, the set of difference \begin{align*} \left\{ \min \{ n \in \mathcal{C} \mid n >m \} -m \mid m \in \mathcal{C} \right \} \end{align*} be bounded?

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(0,2), C=(2,0), D=(2,2), E=(0,1), F=(1,0); draw(A--E--B--F--C--D--A--F^^E--D--B); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, W); label("F", F, S); label("H", (.8,0.6)); label("I", (0.4,1.4)); [/asy] $ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{7}{15}\qquad\textbf{(D)}\ \frac{8}{15}\qquad\textbf{(E)}\ \frac{3}{5} $

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

[u]Set 2[/u] [b]p4.[/b] $\vartriangle ABC$ is an isosceles triangle with $AB = BC$. Additionally, there is $D$ on $BC$ with $AC = DA = BD = 1$. Find the perimeter of $\vartriangle ABC$. [b]p5[/b]. Let $r$ be the positive solution to the equation $100r^2 + 2r - 1 = 0$. For an appropriate $A$, the infinite series $Ar + Ar^2 + Ar^3 + Ar^4 +...$ has sum $1$. Find $A$. [b]p6.[/b] Let $N(k)$ denote the number of real solutions to the equation $x^4 -x^2 = k$. As $k$ ranges from $-\infty$ to $\infty$, the value of $N(k)$ changes only a finite number of times. Write the sequence of values of $N(k)$ as an ordered tuple (i.e. if $N(k)$ went from $1$ to $3$ to $2$, you would write $(1, 3, 2)$). PS. You should use hide for answers.

A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?

How many ordered integer pairs $ \left(x,y\right)$ are there satisfying following equation: \[ y \equal{} \sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1997\plus{}\sqrt{x\plus{}1997\plus{}\ldots \plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}\sqrt{x} } } } } } } }.\] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 1998 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ \text{Infinitely many}$

$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$

Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences)

$a_1,a_2, \cdots , a_{2003}$ are integers such that $|a_1| = 1$ and $|a_{i+1}|=|a_i+1|$ $(1\leq i \leq 2002)$. What is the minimal value of $|a_1+a_2+\cdots + a_{2003}|$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 34 \qquad\textbf{(C)}\ 56 \qquad\textbf{(D)}\ 65 \qquad\textbf{(E)}\ \text{None of the preceding} $