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?

What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$? $ \textbf{a)}\ 1680 \qquad\textbf{b)}\ 882 \qquad\textbf{c)}\ 729 \qquad\textbf{d)}\ 450 \qquad\textbf{e)}\ 246 $

(a) Let there be given a rectangular parallelepiped. Show that some four of its vertices determine a tetrahedron whose all faces are right triangles. (b) Conversely, prove that every tetrahedron whose all faces are right triangles can be obtained by selecting four vertices of a rectangular parallelepiped. (c) Now investigate such tetrahedra which also have at least two isosceles faces. Given the length $a$ of the shortest edge, compute the lengths of the other edges.

Evaluate the following limit. \[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]

Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers, $0$ and $1$. Prove that it is possible to obtain : a) $1/5$, b) an arbitrary rational number between $0$ and $1$.