[b]p1.[/b] For what integers $n$ is $2^6 + 2^9 + 2^n$ the square of an integer? [b]p2.[/b] Two integers are chosen at random (independently, with repetition allowed) from the set $\{1,2,3,...,N\}$. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd. [b]p3.[/b] Let $X$ be a point in the second quadrant of the plane and let $Y$ be a point in the first quadrant. Locate the point $M$ on the $x$-axis such that the angle $XM$ makes with the negative end of the $x$-axis is twice the angle $YM$ makes with the positive end of the $x$-axis. [b]p4.[/b] Let $a,b$ be positive integers such that $a \ge b \sqrt3$. Let $\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3$ for $n = 1,2,3,...$. i. Prove that $\lim_{n \to + \infty} \frac{a_n}{b_n}$ exists. ii. Evaluate this limit. [b]p5.[/b] Suppose $m$ and $n$ are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers $a,b,c,d$, so that $m^2 = a^2 + b^2$ and $n^2= c^2 + d^2$. Show than $mn$ is the hypotenuse of at least two distinct Pythagorean triangles. Hint: you may not assume that the pair $(a,b)$ is different from the pair $(c,d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b,c,d$ be four positive real numbers. If they satisfy \[a+b+\frac{1}{ab}=c+d+\frac{1}{cd}\quad\text{and}\quad\frac1a+\frac1b+ab=\frac1c+\frac1d+cd\] then prove that at least two of the values $a,b,c,d$ are equal.

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?

$100$ points are marked in the plane so that no three of marked points are collinear. One of marked points is red, and the others are blue. A triangle with vertices at blue points is called [i]good[/i] if the red point lies inside it. Determine if it is possible that the number of good triangles is not less than the half of the total number of traingles with vertices at blue points.

If $\sum_{i=1}^{n} \cos ^{-1}\left(\alpha_{i}\right)=0,$ then find $\sum_{i=1}^{n} \alpha_{i}$ [list=1] [*] $\frac{n}{2} $ [*] $n $ [*] $n\pi $ [*] $\frac{n\pi}{2} $ [/list]

In the Faculty of Cybernetics football championship, \( n \geq 3 \) teams participated. The competition was held in a round-robin format, meaning that each team played against every other team exactly once. For a win, a team earns 3 points, for a loss no points are awarded, and for a draw, both teams receive 1 point each. It turned out that the winning team scored strictly more points than any other team and had at most as many wins as losses. What is the smallest \( n \) for which this could happen? [i]Proposed by Bogdan Rublov[/i]

Determine all pairs of polynomials $f$ and $g$ with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]

What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$? (A) $ -1$ (B) $ -2$ (C) $-3$ (D) $-4$ (E) None of the above

Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.

Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$-gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$, $i=1,2,\ldots,n$. Prove the equality $$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$

A jeweler makes necklaces with round stones, four emeralds (green) and four rubies (red), arranged at equal distances from each other. One day, they decide to give away some necklaces. How many necklaces can they give away without the risk of two friends ending up with the same necklace? (*Observation: The necklace is completely symmetrical except for the type of stone, meaning there is not a unique way to form it. Consider this while solving the problem.*)

The plane is colored with two colors so that the following property holds: for each real $a>0$ there is an equilateral triangle of side length $a$ whose $3$ vertices are of the same color. Prove that for any three numbers $a,b,c>0$ for which the sum of any two is greater than the third there is a triangle with sides $a$, $b$, and $c$ whose $3$ vertices are of the same color.

Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively. Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.

For every positive integer $M \geq 2$, find the smallest real number $C_M$ such that for any integers $a_1, a_2,\ldots , a_{2023}$, there always exist some integer $1 \leq k < M$ such that  \[\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.\] Here, $\{x\}$ is the unique number in the interval $[0, 1)$ such that $x - \{x\}$ is an integer. [i] Proposed by usjl[/i]

Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$. The Editors

$b$ and $c$ are positive. Prove the inequality \[ \left(b-c\right)^{2011}\left(b+c\right)^{2011}\left(c-b\right)^{2011} \geq \left(b^{2011}-c^{2011}\right)\left(b^{2011}+c^{2011}\right)\left(c^{2011}-b^{2011}\right). \] (Author: V. Senderov)

A square grid $2n \times 2n$ is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of 3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.

For three positive real numbers $ a,b,c $ satisfying the condition $ \frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca} =1, $ prove that $$ 3/2\le \frac{ab-1}{ab+1} +\frac{bc-1}{bc+1} +\frac{ca-1}{ca+1} <2. $$ [i]Mircea Becheanu[/i]

Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?

On side \( AC \) of triangle \( ABC \), a point \( P \) is chosen such that \( AP = \frac{1}{3} AC \), and on segment \( BP \), a point \( S \) is chosen such that \( CS \perp BP \). A point \( T \) is such that \( BCST \) is a parallelogram. Prove that \( AB = AT \). [i]Proposed by Bohdan Zheliabovskyi[/i]

Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$

Let $n = (p^2 +2)^2 -9(p^2 -7)$ where $p$ is a prime number. Determine the smallest value of the sum of the digits of $n$ and for what prime number $p$ is obtained.

Let $C_1, A_1, B_1$ be points on sides $AB, BC, CA$ of triangle $ABC$, such that $AA_1, BB_1, CC_1$ concur. The rays $B_1A_1$ and $B_1C_1$ meet the circumcircle of the triangle at points $A_2$ and $C_2$ respectively. Prove that $A, C$, the common point of $A_2C_2$ and $BB_1$ and the midpoint of $A_2C_2$ are concyclic.

Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine: a) which vertex of $BCF$ is its apex, b) the size of $\angle BAC$