Prove that the sequence defined by: $$ y_ {n + 1} = \frac {1} {2} (3y_ {n} + \sqrt {5y_ {n} ^ {2} -4}) , \,\, \forall n \ge 0$$ with $ y_ {0} = 1$ consists only of integers.

Given an acute-angled triangle $ABC$. A straight line parallel to $BC$ intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At what location of the points $M$ and $P$ will the radius of the circle circumscribed about the triangle $BMP$ be the smallest? (I. Sharygin)

Let $f\left(x\right)$ be the greatest prime number at most $x$. Let $g\left(x\right)$ be the least prime number greater than $x$. Find $$\sum_{i=2}^{100}\frac{1}{f\left(i\right)g\left(i\right)}.$$ [i]2017 CCA Math Bonanza Lightning Round #4.3[/i]

We are given a disk $\mathcal D$ of radius $1$ in the plane. (a) Prove that $\mathcal D$ cannot be covered with two disks of radii $r<1$. (b) Prove that, for some $r<1$, $\mathcal D$ can be covered with three disks of radius $r$. What is the smallest such $r$?

Consider a finite field $ K. $ [b]a)[/b] Prove that there is an element $ k $ in $ K $ having the property that the polynom $ X^3+k $ is irreducible in $ K[X], $ if $ \text{ord} (K)\equiv 1\pmod {12}. $ [b]b)[/b] Is [b]a)[/b] still true if, intead, $ \text{ord} (K) \equiv -1\pmod{12} ? $ [i]Dorel Miheț[/i]

Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$. a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$. b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.

$16.$ In an acute angle triangle $ABC,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $BC.$ Let $F$ be the midpoint of $AC.$ Suppose $\angle{BAE}=40^o. $ If $\angle{DAE}=\angle{DFE},$ What is the magnitude of $\angle{ADF}$ in degrees $?$

On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions: (1) Each block must have at most $50$ states. (2) Every pair of states must be together in at least one block. Find the minimum number of blocks that must be formed.

Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime. [i]Evan o'Dorney[/i]

Given is an integer sequence $\{a_n\}_{n \ge 0}$ such that $a_{0}=2$, $a_{1}=3$ and, for all positive integers $n \ge 1$, $a_{n+1}=2a_{n-1}$ or $a_{n+1}= 3a_{n} - 2a_{n-1}$. Does there exist a positive integer $k$ such that $1600 < a_{k} < 2000$?

Prove that exists sequences $ (a_n)_{n\ge 0}$ with $ a_n\in \{\minus{}1,\plus{}1\}$, for any $ n\in \mathbb{N}$, such that: \[ \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}\]

Asheville, Bakersfield, Charter, and Darlington are four small towns along a straight road in that order. The distance from Bakersfield to Charter is one-third the distance from Asheville to Charter and one-quarter the distance from Bakersfield to Darlington. If it is $12$ miles from Bakersfield to Charter, how many miles is it from Asheville to Darlington?

A chord $PQ$ of the circumcircle of a triangle $ABC$ meets the sides $BC, AC$ at points $A', B'$ respectively. The tangents to the circumcircle at $A$ and $B$ meet at a point $X$, and the tangents at points $P$ and $Q$ meet at point $Y$. The line $XY$ meets $AB$ at a point $C'$. Prove that the lines $AA', BB'$ and $CC'$ concur.

A straight line joins the points $ (\minus{}1,1)$ and $ (3,9)$. Its $ x$-intercept is: $ \textbf{(A)}\ \minus{}\frac{3}{2}\qquad \textbf{(B)}\ \minus{}\frac{2}{3}\qquad \textbf{(C)}\ \frac{2}{5}\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 3$

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the [i]Primle[/i]. For each guess, a digit is highlighted blue if it is in the [i]Primle[/i], but not in the correct place. A digit is highlighted orange if it is in the [i]Primle[/i] and is in the correct place. Finally, a digit is left unhighlighted if it is not in the [i]Primle[/i]. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the [i]Primle[/i]? $$\begin{array}{c} \boxed{1} \,\, \boxed{3} \\[\smallskipamount] \boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7} \end{array}$$ [i]Proposed by Andrew Wu and Jason Wang[/i]

A Mediterranean polynomial has only real roots and it is of the form \[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

Consider a $10 \times 10$ grid. On every move, we colour $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?

Peter and Bob play a game on a $n\times n$ chessboard. At the beginning, all squares are white apart from one black corner square containing a rook. Players take turns to move the rook to a white square and recolour the square black. The player who can not move loses. Peter goes first. Who has a winning strategy?

Let $a,b,c$ be positive integers. Prove that $a^2+b^2+c^2$ is divisible by $4$, if and only if $a,b,c$ are even.

If $a$ and $b$ are positive integers, we say that $a$ [i]almost divides[/i] $b$ if $a$ divides at least one of $b - 1$ and $b + 1$. We call a positive integer $n$ [i]almost prime[/i] if the following holds: for any positive integers $a, b$ such that $n$ almost divides $ab$, we have that $n$ almost divides at least one of $a$ and $b$. Determine all almost prime numbers. [hide = original link][url]https://mathematical.olympiad.ch/fileadmin/user_upload/Archiv/Intranet/Olympiads/Mathematics/deploy/exams/2024/FinalRound/Exam/englishFinalRound2024.pdf[/url]!![/hide]

In a closed rectangular neighborhood there are: $S$ streets (these are straight roads of maximum length), $V$ four-arm intersections ( [img]https://cdn.artofproblemsolving.com/attachments/e/4/6a5974a30dc182b59a519a8ef4eb4f1412e05e.png[/img]), $H$ city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and $T$ represents the number of $T$-intersections ([img]https://cdn.artofproblemsolving.com/attachments/0/a/b390a30a0b27d83db681f70f633bdeed697163.png[/img] ). For example, in the neighborhood below, there are $15$ streets, $8$ four-arm intersections, $20$ city blocks and $22$ $T$-intersections. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c1a5e463d0fb5671ac0702c91cfc2272d4e2c3.png[/img] Prove that in each district $S + V = H + 3$.

Let $ABC$ be a triangle with $O$ as its circumcenter. A circle $\Gamma$ tangents $OB, OC$ at $B, C$, respectively. Let $D$ be a point on $\Gamma$ other than $B$ with $CB=CD$, $E$ be the second intersection of $DO$ and $\Gamma$, and $F$ be the second intersection of $EA$ and $\Gamma$. Let $X$ be a point on the line $AC$ so that $XB\perp BD$. Show that one half of $\angle ADF$ is equal to one of $\angle BDX$ and $\angle BXD$. [i]Proposed by usjl[/i]

Given two regular tetrahedrons with edges of length $\sqrt2$, transforming into one another with central symmetry. Let $\Phi$ be the set the midpoints of segments whose ends belong to different tetrahedrons. Find the volume of the figure $\Phi$.