Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

Let \( \mathbb{R} \) denote the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \[ f(x^2) - y f(y) = f(x+y)(f(x) - y) \] for all real numbers \( x \) and \( y \).

Let $S$ be the set of integers that can be written in the form $50m + 3n$ where $m$ and $n$ are non-negative integers. For example $3, 50, 53$ are all in $S$. Find the sum of all positive integers not in $S$.

Show that the GCD of three consecutive triangular numbers is 1.

Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.

We divide a convex $2009$-gon in triangles using non-intersecting diagonals. One of these diagonals is colored green. It is allowed the following operation: for two triangles $ABC$ and $BCD$ from the dividing/separating with a common side $BC$ if the replaced diagonal was green it loses its color and the replacing diagonal becomes green colored. Prove that if we choose any diagonal in advance it can be colored in green after applying the operation described finite number of times.

Prove that: $\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}=\sqrt{2}+\sqrt3+\sqrt5$

Let $ABC$ be a triangle with no right angle, $E$ on the line $BC$ such that $\angle AEB = \angle BAC$ and $\Delta_A$ the perpendicular to $BC$ at $E$. Let the circle $\gamma$ with diameter $BC$ intersect $BA$ again at $D$. For each point $M$ on $\gamma$ ($M$ is distinct from $B$), the line $BM$ meets $\Delta_A$ at $M'$ and the line $AM$ meets $\gamma$ again at $M''$. (a) Show that $p(A) = AM' \times DM''$ is independent of the chosen $M$. (b) Keeping $B,C$ fixed, and let $A$ vary. Show that $\frac{p(A)}{d(A,\Delta_A)}$ is independent of $A$. Michel Bataille

Given $n$ points, some of them connected by non-intersecting segments. You can reach every point from every one, moving along the segments, and there is no couple, connected by two different ways. Prove that the total number of the segments is $(n-1)$.

In a factory, square tables of $ 40 \times 40$ are tiled with four tiles of size $ 20 \times 20$. All tiles are the same and decorated in the same way with an asymmetric pattern such as the letter $ J$. How many different types of tables can be produced in this way?

In a triangle $ABC$, $\angle B = 70^o$, $\angle C = 50^o$. A point $M$ is taken on the side $AB$ such that $\angle MCB = 40^o$ , and a point $N$ is taken on the side $AC$ such that $\angle NBC = 50^o$. Find $\angle NMC$.

Let $M$ be a map made of several cities linked to each other by flights. We say that there is a route between two cities if there is a nonstop flight linking these two cities. For each city to the $M$ denote by $M_a$ the map formed by the cities that have a route to and routes linking these cities together ( to not part of $M_a$). The cities of $M_a$ are divided into two sets so that the number of routes linking cities of different sets is maximum; we call this number the cut of $M_a$. Suppose that for every cut of $M_a$ , it is strictly less than two thirds of the number of routes $M_a$ . Show that for any coloring of the $M$ routes with two colors there are three cities of $M$ joined by three routes of the same color.

A is a two-digit number and B is a three-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).

Let a_1, a_2, a_3,... be a sequence of positive numbers. If there exists a positive number M such that for n = 1,2,3,..., $a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n}< Ma^{2}_{n+1}$ then prove that there exist a positive number M' such that for every n = 1,2,3,..., $a_{1}+a_{2}+...+a_{n}< M'a_{n+1}$

Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and let $M$ be the midpoint of $BC$. Points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that $MPQ$ and $ABC$ are similar (with vertices in that order). The product of all different possible areas of $MPQ$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]2021 CCA Math Bonanza Lightning Round #4.4[/i]

Let $A$ be an $n\times n$ complex matrix such that $A \ne \lambda I_{n}$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the maindiagonal.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$

Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.

Let $a_1, a_2, \ldots, a_{2023}$ be positive reals such that $\sum_{i=1}^{2023}a_i^i=2023$. Show that $$\sum_{i=1}^{2023}a_i^{2024-i}>1+\frac{1}{2023}.$$

Does there exist a $4$-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of $1992$?

Let $n$ be a positive integer. Determine the smallest possible value of $1-n+n^2-n^3+\dots+n^{1000}$. [i]Proposed by Evan Chen[/i]

Let $a$ and $b$ be randomly selected three-digit integers and suppose $a > b$. We say that $a$ is [i]clearly bigger[/i] than $b$ if each digit of $a$ is larger than the corresponding digit of $b$. If the probability that $a$ is clearly bigger than $b$ is $\tfrac mn$, where $m$ and $n$ are relatively prime integers, compute $m+n$. [i]Proposed by Evan Chen[/i]

Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$ Prove that $C,J,H,I$ are concyclic.

Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$. Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.

In a town there are $n$ avenues running from south to north. They are numbered $1$ through $n$ (from west to east). There are $n$ streets running from west to east – they are also numbered $1$ through $n$ (from south to north). If you drive through the junction of the $k$th avenue and the $\ell$th street, you have to pay $k\ell$ kroner. How much do you at least have to pay for driving from the junction of the $1$st avenue and the $1$st street to the junction of the nth avenue and the $n$th street? (You also pay for the starting and finishing junctions.)