Determine all integers $n$ such that both of the numbers: $$|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|$$ are both prime numbers.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then a)$800000000$; b)$880000000$ by such replacements?

Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$. Find the radius $r$.

Let $A, B \in \mathbb{N}$. Consider a sequence $x_1, x_2, \ldots$ such that for all $n\geq 2$, $$x_{n+1}=A \cdot \gcd(x_n, x_{n-1})+B. $$ Show that the sequence attains only finitely many distinct values.

Let $ m,n$ be positive integers. Prove that the number $ 5^n\plus{}5^m$ can be represented as sum of two perfect squares if and only if $ n\minus{}m$ is even. [i]Vasile Zidaru[/i]

For all positive integers $n$, show that: $$ \dfrac1n \sum^n _{k=1} \dfrac{k \cdot k! \cdot {n\choose k}}{n^k} = 1$$

Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.

Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set $$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and $$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.

Let $ABC$ be a right-angled triangle with $\angle C = 90^{\circ}$, $K$, $L$, $M$ be the midpoints of sides $AB$, $BC$, $CA$ respectively, and $N$ be a point of side $AB$. The line $CN$ meets $KM$ and $KL$ at points $P$ and $Q$ respectively. Points $S$, $T$ lying on $AC$ and $BC$ respectively are such that $APQS$ and $BPQT$ are cyclic quadrilaterals. Prove that a) if $CN$ is a bisector, then $CN$, $ML$ and $ST$ concur; b) if $CN$ is an altitude, then $ST$ bisects $ML$.

Let $p$ be the third-smallest prime number greater than $5$ such that: $\bullet$ $2p + 1$ is prime, and $\bullet$ $5^p \not\equiv 1$ (mod $2p + 1$). Find $p$.

If $ \left(r \plus{} \frac {1}{r}\right)^2 \equal{} 3$, then $ r^3 \plus{} \frac {1}{r^3}$ equals $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!

Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $ [i]Florian Dumitrel[/i]

Let $M$ be a centre of gravity (the intersection point of the medians) of a triangle $ABC$. Under rotation by $120$ degrees about the point $M$, the point $B$ is taken to the point $P$; under rotation by $240$ degrees about $M$, the point $C$ is taken to the point $Q$. Prove that either $APQ$ is an equilateral triangle, or the points $A, P, Q$ coincide. (Bykovsky, Khabarovsksk)

Asteroids A and B have circular orbits around the same star. Asteroid A is located 400 km away from the star and takes 8000 hours to complete one full revolution. Asteroid B is located 100 km away and the speed of Asteroid B is twice the speed of Asteroid A. Find how long it takes for Asteroid B to complete one full revolution in hours. [i]2022 CCA Math Bonanza Individual Round #1[/i]

Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black. Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

A; B; C; D are the vertices of a square, and P is a point on the arc CD of its circumcircle. Prove that $ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $ Can anyone here find the solution? I'm not great with geometry, so i tried turning it into co-ordinate geometry equations, but sadly to no avail. Thanks in advance.

Misha is currently taking a Complexity Theory exam, but he seems to have forgotten a lot of the material! In the question, he is asked to fill in the following boxes with $\subseteq$ and $\subsetneq$ to identify the relationship between different complexity classes: $$\mathsf{NL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{NP}\ \fbox{\phantom{tt}}\ \mathsf{PH}\ \fbox{\phantom{tt}}\ \mathsf{PSPACE}\ \fbox{\phantom{tt}}\ \mathsf {EXP}$$ and $$\mathsf{coNL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{coNP}\ \fbox{\phantom{tt}}\ \mathsf{PH}$$ Luckily, he remembers that $\mathsf{P} \neq \mathsf{EXP}$, $\mathsf{NL} \neq \mathsf{PSPACE}$, $\mathsf{coNL} \neq \mathsf{PSPACE}$, and $\mathsf{NP} \neq \mathsf{coNP}\implies \mathsf{P}\neq \mathsf{NP} \land \mathsf{P}\neq \mathsf{coNP}$. How many ways are there for him to fill in the boxes so as not to contradict what he remembers?

Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is defined as follows: we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.

Compute the number of distinct pairs of the form \[(\text{first three digits of }x,\text{ first three digits of }x^4)\] over all integers $x>10^{10}$. For example, one such pair is $(100,100)$ when $x=10^{10^{10}}$.

How many pairs of positive integers $(x,y)$ are there such that $\sqrt{xy}-71\sqrt x + 30 = 0$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 2130 \qquad\textbf{(E)}\ \text{Infinitely many} $

Let $a,b,c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge \frac12 (1+abc)$$

Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.

Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying \[ a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n} \] for all $n\geq 2024^{2024}$ is bounded. [i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$