Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

The Fibonacci numbers $F_1, F_2, F_3, \ldots$ are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for each integer $n \ge 1$. Let $P$ be the unique polynomial of least degree for which $P(n) = F_n$ for all integers $1 \le n \le 10$. Compute the integer $m$ for which \[P(100) - \sum_{k=11}^{98} P(k) = \frac{m}{10} \dbinom{98}{9} + 144.\] [i]Proposed by Michael Tang[/i]

Find the smallest real number $M$ such that $$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$ for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)

Triangle $ABC$ has a right angle at $C$. Let $D$ be the midpoint of side $\overline{AC}$, and let $E$ be the intersection of $\overline{AC}$ and the bisector of $\angle ABC$. The area of $\triangle ABC$ is $144$, and the area of $\triangle DBE$ is $8$. Find $AB^2$.

A block of mass $m_\text{1}$ is on top of a block of mass $m_\text{2}$. The lower block is on a horizontal surface, and a rope can pull horizontally on the lower block. The coefficient of kinetic friction for all surfaces is $\mu$. What is the resulting acceleration of the lower block if a force $F$ is applied to the rope? Assume that $F$ is sufficiently large so that the top block slips on the lower block. [asy] size(200); import roundedpath; draw((0,0)--(30,0),linewidth(3)); path A=(7,0.5)--(17,0.5)--(17,5.5)--(7,5.5)--cycle; filldraw(roundedpath(A,1),lightgray); path B=(10,6)--(15,6)--(15,9)--(10,9)--cycle; filldraw(roundedpath(B,1),lightgray); label("1",(12.5,6),1.5*N); label("2",(12,0.5),3*N); draw((17,3)--(27,3),EndArrow(size=13)); label(scale(1.2)*"$F$",(22,3),2*N); [/asy] (A) $a_\text{2}=(F-\mu g(2m_\text{1}+m_\text{2}))/m_\text{2}$ (B) $a_\text{2}=(F-\mu g(m_\text{1}+m_\text{2}))/m_\text{2}$ (C) $a_\text{2}=(F-\mu g(m_\text{1}+2m_\text{2}))/m_\text{2}$ (D) $a_\text{2}=(F+\mu g(m_\text{1}+m_\text{2}))/m_\text{2}$ (E) $a_\text{2}=(F-\mu g(m_\text{2}-m_\text{1}))/m_\text{2}$

How many non-congruent triangles have vertices at three of the eight points in the array shown below? [asy]dot((0,0)); dot((0,.5)); dot((.5,0)); dot((.5,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));[/asy] $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

$5000$ movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the first kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).

Let $ABC$ be a scalene acute triangle with circumcenter $O$. Let $K$ be a point on the side $\overline{BC}$. Define $M$ as the second intersection of $\overleftrightarrow{OK}$ with the circumcircle of $BOC$. Let $L$ be the reflection of $K$ by $\overleftrightarrow{AC}$. Show that the circumcircles of the triangles $LCM$ and $ABC$ are tangent if, and only if, $\overline{AK} \perp \overline{BC}$.

Find all triples ( $\alpha, \beta,\theta$) of acute angles such that the following inequalities are fulfilled at the same time $$(\sin \alpha + \cos \beta + 1)^2 \ge 2(\sin \alpha + 1)(\cos \beta + 1)$$ $$(\sin \beta + \cos \theta + 1)^2 \ge 2(\sin \beta + 1)(\cos \theta + 1)$$ $$(\sin \theta + \cos \alpha + 1)^2 \ge 2(\sin \theta + 1)(\cos \alpha + 1).$$

Max flips $2020$ fair coins. Let the probability that there are at most $505$ heads be $p$. Estimate $-\log_2(p)$ to 5 decimal places, in the form $x.abcde$ where $x$ is a positive integer and $a, b, c, d, e$ are decimal digits.

[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \] [b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)} \] Calculate the minimum value of $k.$

For how many triples of positive integers $(a,b,c)$ with $1\leq a,b,c\leq 5$ is the quantity \[(a+b)(a+c)(b+c)\] not divisible by $4$?

Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?

The roots of the equation $x^2+5x-7=0$ are $x_1$ and $x_2$. What is the value of $x_1^3+5x_1^2-4x_1+x_1^2x_2-4x_2$ ? $\textbf{(A)}\ -15 \qquad\textbf{(B)}\ 175+25\sqrt{53} \qquad\textbf{(C)}\ -50 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ \text{None}$

Let $V = \left\{ 1, \dots, 8 \right\}$. How many permutations $\sigma : V \to V$ are automorphisms of some tree? (A $\emph{graph}$ consists of some set of vertices and some edges between pairs of distinct vertices. It is $\emph{connected}$ if every two vertices in it are connected by some path of one or more edges. A $\emph{tree}$ $G$ on $V$ is a connected graph with vertex set $V$ and exactly $|V|-1$ edges, and an $\emph{automorphism}$ of $G$ is a permutation $\sigma : V \to V$ such that vertices $i,j \in V$ are connected by an edge if and only if $\sigma(i)$ and $\sigma(j)$ are.)

Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\, b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$ Prove that $$|a-b| <\frac{1}{99! 100!}$$ (G Galperin, Moscow)

Let $A,B,C,D$ be points on the line $d$ in that order and $AB = CD$. Denote $(P)$ as some circle that passes through $A, B$ with its tangent lines at $A, B$ are $a,b$. Denote $(Q)$ as some circle that passes through $C, D$ with its tangent lines at $C, D$ are $c,d$. Suppose that $a$ cuts $c, d$ at $K, L$ respectively and $b$ cuts $c, d$ at $M, N$ respectively. Prove that four points $K, L, M,N$ belong to a same circle $(\omega)$ and the common external tangent lines of circles $(P)$, $(Q)$ meet on $(\omega)$.

Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin $2013$ times. On the $n^{\text{th}}$ flip (where $n=1,2,\dots,2013$), Ben does the following if the coin flips heads: (i) If the blackboard is empty, Ben writes $n$ on the blackboard. (ii) If the blackboard is not empty, let $m$ denote the largest number on the blackboard. If $m^2+2n^2$ is divisible by $3$, Ben erases $m$ from the blackboard; otherwise, he writes the number $n$. No action is taken when the coin flips tails. If probability that the blackboard is empty after all $2013$ flips is $\frac{2u+1}{2^k(2v+1)}$, where $u$, $v$, and $k$ are nonnegative integers, compute $k$. [i]Proposed by Evan Chen[/i]

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.

Let $n\in \Bbb{N}, n \geq 4.$ Determine all sets $ A = \{a_1, a_2, . . . , a_n\} \subset \Bbb{N}$ containing $2015$ and having the property that $ |a_i - a_j|$ is prime, for all distinct $i, j\in \{1, 2, . . . , n\}.$

Higher Secondary P2 Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.

Prove that $\frac1{2002}<\frac12\cdot\frac34\cdot\frac56\cdots\frac{2001}{2002}<\frac1{44}$.

Let $\theta$ be a positive real number. Show that if $k\in \mathbb{N}$ and both $\cosh k \theta$ and $\cosh(k+1) \theta$ are rational, then so is $\cosh \theta$.