Let $f: \mathbb{R} \to \mathbb{R}$ be a non-decreasing function such that $f(y) - f(x) < y - x$ for all real numbers $x$ and $y > x$. The sequence $u_1,u_2,\ldots$ of real numbers is such that $u_{n+2} = f(u_{n+1}) - f(u_n)$ for all $n\geq 1$. Prove that for any $\varepsilon > 0$ there exists a positive integer $N$ such that $|u_n| < \varepsilon$ for all $n\geq N$.

A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry. [i]Proposed by Ethan Tan[/i]

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$. a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$. b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

Let $A$ and $B$ be two distinct points in the plane. Let $M$ be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$. Prove that the lines $AT$ and $XY$ are parallel. [i]Navneel Singhal, India[/i]

Let $D, E$ and $F{}$ be the midpoints of the sides $BC, CA$ and $AB{}$ of the acute-angled triangle $ABC$ and let $H_a, H_b$ and $H_c{}$ be the orthocenters of the triangles $ABD, BCE$ and $CAF{}$ respectively. Prove that the triangles $H_aH_bH_c$ and $DEF$ have equal areas. [i]Proposed by Tran Quang Hung[/i]

[b]2/1/27.[/b] Suppose $a, b,$ and $c$ are distinct positive real numbers such that \begin{align*}abc=1000, \\ bc(1-a)+a(b+c)=110.\end{align*} If $a<1$, show that $10<c<100$.

For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$, and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$. [i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]

Prove that for all positive integers $n$, $1^3 + 2^3 + 3^3 +\dots+n^3$ is a perfect square.

Sides of a triangle are equal to $3$, $4$ and $5$. Each side is extended until it intersects the bisector of the external angle to the angle opposite to it. Three such points are obtained in all. Prove that one of the three points we get is the midpoint of the segment joining the other two points. (V. Prasolov)

Let $c$ be the length of the hypotenuse of a right angle triangle whose two other sides have lengths $a$ and $b$. Prove that $a+b\le c\sqrt{2}$. When does the equality hold?

Let $n> 1$ be an odd integer. On a square surface have been placed $n^2 - 1$ white slabs and a black slab on the center. Two workers $A$ and $B$ take turns removing them, betting that whoever removes black will lose. First $A$ picks a slab; if it has row number $i \ge (n + 1) / 2$, then it will remove all tiles from rows with number greater than or equal to$ i$, while if $i <(n + 1) / 2$, then it will remove all tiles from the rows with lesser number or equal to $i$. Proceed in a similar way with columns. Then $B$ chooses one of the remaining tiles and repeats the process. Determine who has a winning strategy and describe it. Note: Row and column numbering is ascending from top to bottom and from left to right.

Show that if the series $a_1 + a_2 + a_3 + \cdots + a_n + \cdots$ converges, then the series $a_1 + a_2 / 2 + a_3 / 3 + \cdots + a_n / n + \cdots$ converges also.

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

Consider a $2024 \times 2024$ grid of unit squares. Two distinct unit squares are adjacent if they share a common side. Each unit square is to be coloured either black or white. Such a colouring is called $\textit{evenish}$ if every unit square in the grid is adjacent to an even number of black unit squares. Determine the number of $\textit{evenish}$ colourings.