Find all pair of positive integers $(x, y)$ satisfying the equation \[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

A special kind of chess knight is in the origin of an infinite grid. It can make one of twelve different moves: it can move directly up, down, left, or right one unit square, or it can move $1$ units in one direction and $3$ units in an orthogonal direction. How many different squares can it be on after $2$ moves?

Consider $P$ a regular $9$-sided convex polygon with each side of length $1$. A diagonal at $P$ is any line joining two non-adjacent vertices of $P$. Calculate the difference between the lengths of the largest and smallest diagonal of $P$.

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

If $x, y$ and $z$ are real numbers such that $x^2 + y^2 + z^2 = 2$, prove that $x + y + z \le xyz + 2$.

Find the last digit of \[7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.\]

There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ? [i](i)[/i] Every route includes exactly three ports; [i](ii)[/i] No two routes contain the same three ports; [i](iii)[/i] The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.

An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle in point $M$. Find the locus of circumcenters of triangles $AMK$.

Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.

Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.

Let $ z_1,z_2,z_3 $ be nonzero complex numbers and pairwise distinct, having the property that $\left( z_1+z_2\right)^3 =\left( z_2+z_3\right)^3 =\left( z_3+z_1\right)^3. $ Show that $ \left| z_1-z_2\right| =\left| z_2-z_3\right| =\left| z_3-z_1\right| . $

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Given some $m\times n$ table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.

We define $\mathbb F_{101}[x]$ as the set of all polynomials in $x$ with coefficients in $\mathbb F_{101}$ (the integers modulo $101$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb F_{101}$ for each nonnegative integer $k$. For example, $(x+3)(100x+5)=100x^2+2x+15$ in $\mathbb F_{101}[x]$ because the corresponding coefficients are equal modulo $101$. We say that $f(x)\in\mathbb F_{101}[x]$ is \emph{lucky} if it has degree at most $1000$ and there exist $g(x),h(x)\in\mathbb F_{101}[x]$ such that \[f(x)=g(x)(x^{1001}-1)+h(x)^{101}-h(x)\] in $\mathbb F_{101}[x]$. Find the number of lucky polynomials. [i]Proposed by Michael Ren.[/i]

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power: $$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$ [i]Proposed by Navid Safaei[/i]

In $\Delta ABC$, $AB=6$, $BC=7$ and $CA=8$. Let $D$ be the mid-point of minor arc $AB$ on the circumcircle of $\Delta ABC$. Find $AD^2$

Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides? [asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("$10$",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("$252^{\circ}$",(0.05,0.05),NE);[/asy] [asy] import three; picture mainframe; defaultpen(fontsize(11pt)); picture conePic(picture pic, real r, real h, real sh) { size(pic, 3cm); triple eye = (11, 0, 5); currentprojection = perspective(eye); real R = 1, y = 2; triple center = (0, 0, 0); triple radPt = (0, R, 0); triple negRadPt = (0, -R, 0); triple heightPt = (0, 0, y); draw(pic, arc(center, radPt, negRadPt, heightPt, CW)); draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8")); draw(pic, center--radPt, linetype("8 8")); draw(pic, center--heightPt, linetype("8 8")); draw(pic, negRadPt--heightPt--radPt); label(pic, (string) r, center--radPt, dir(270)); if (h != 0) { label(pic, (string) h, heightPt--center, dir(0)); } if (sh != 0) { label(pic, (string) sh, heightPt--radPt, dir(0)); } return pic; } picture pic1; pic1 = conePic(pic1, 6, 0, 10); picture pic2; pic2 = conePic(pic2, 6, 10, 0); picture pic3; pic3 = conePic(pic3, 7, 0, 10); picture pic4; pic4 = conePic(pic4, 7, 10, 0); picture pic5; pic5 = conePic(pic5, 8, 0, 10); picture aux1; picture aux2; picture aux3; add(aux1, pic1.fit(), (0,0), W); label(aux1, "$\textbf{(A)}$", (0,0), 22W, linewidth(4)); label(aux1, "$\textbf{(B)}$", (0,0), 3E); add(aux1, pic2.fit(), (0,0), 35E); add(aux2, aux1.fit(), (0,0), W); label(aux2, "$\textbf{(C)}$", (0,0), 3E); add(aux2, pic3.fit(), (0,0), 35E); add(aux3, aux2.fit(), (0,0), W); label(aux3, "$\textbf{(D)}$", (0,0), 3E); add(aux3, pic4.fit(), (0,0), 35E); add(mainframe, aux3.fit(), (0,0), W); label(mainframe, "$\textbf{(E)}$", (0,0), 3E); add(mainframe, pic5.fit(), (0,0), 35E); add(mainframe.fit(), (0,0), N); [/asy]

Let $p(x)$ be the polynomial with leading coefficent 1 and rational coefficents, such that \[p\left(\sqrt{3 + \sqrt{3 + \sqrt{3 + \ldots}}}\right) = 0,\] and with the least degree among all such polynomials. Find $p(5)$.

Two dice appear to be standard dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Determine all polynomials $P$ with real coefficients satisfying the following condition: whenever $x$ and $y$ are real numbers such that $P(x)$ and $P(y)$ are both rational, so is $P(x + y)$.

Find the number of positive integers $x$ that satisfy \[ \left \lfloor{\frac{2024}{ \left \lfloor \frac{2024}{x} \right \rfloor }} \right \rfloor = x.\]

Find all real numbers $k$ such that $r^4+kr^3+r^2+4kr+16=0$ is true for exactly one real number $r$.

What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take?