For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let \[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & B_{n \minus{} 1} \\ \end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & 0 \\ \end{array} \right). \] Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.

Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.

Given a set of $100$ different numbers such that if each number in the set is replaced by the sum of the others, the same set will be obtained. Prove that the product of numbers in a set is positive.

Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.

Let $S$ be a point inside triangle $ABC$ and let lines $AS$, $BS$ and $CS$ intersect sides $BC$, $CA$ and $AB$ in points $X$, $Y$ and $Z$, respectively. Prove that $$\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1$$ iff $S$ is incenter of $ABC$

Within an equilateral triangle of side $ 15$ are $ 111$ points. Prove that it is always possible to cover three of these points by a round coin of diameter $ \sqrt{3}$, part of which may lie outside the triangle.

Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

In triangle $ABC, \angle C= 60^o, \angle A= 45^o$. Let $M$ be the midpoint of $BC, H$ be the orthocenter of triangle $ABC$. Prove that line $MH$ passes through the midpoint of arc $AB$ of the circumcircle of triangle $ABC$.

The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$ $ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $ $\textbf{(E)}\text{ must be between -1 and +1} $

Before starting to paint, Bill had $ 130$ ounces of blue paint, $ 164$ ounces of red paint, and $ 188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stirpe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.

If $a,b,c\in(0,\infty)$ such that $a+b+c=3$, then $$\frac{a}{3a+bc+12}+\frac{b}{3b+ca+12}+\frac{c}{3c+ab+12}\le \frac{3}{16}$$

Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$. [i]Proposed by Ali Khezeli[/i]

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $? [i]Proposed by Alexander S. Golovanov, Russia[/i]

A triangle is [i]nondegenerate[/i] if its three vertices are not collinear. A particular nondegenerate triangle $\triangle JHU$ has side lengths $x,$ $y,$ and $z,$ and its angle measures, in degrees, are all integers. If there exists a nondegenerate triangle with side lengths $x^2,$ $y^2,$ and $z^2,$ then what is the largest possible angle measure in the original triangle $\triangle JHU,$ in degrees?

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

[b]7.1[/b] A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square. [b]7.2[/b] Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17. [b]7.3 [/b] In a $1000$-digit number, all but one digit is a five. Prove that this number is not a perfect square. [b]7.4 / 6.5[/b] Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]7.5[/b] In a pentagon $ABCDE$, $K$ is the midpoint of $AB$, $L$ is the midpoint of $BC$, $M$ is the midpoint of $CD$, $N$ is the midpoint of $DE$, $P$ is the midpoint of $KM$, $Q$ is the midpoint of $LN$. Prove that the segment $ PQ$ is parallel to side $AE$ and is equal to its quarter. [img]https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png[/img] [b]7.6 / 8.4[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b] A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness [b]One[/b] said exactly one of the four witnesses is a liar. Witness [b]Two[/b] said exactly two of the four witnesses is a liar. Witness [b]Three[/b] said exactly three of the four witnesses is a liar. Witness [b]Four[/b] said exactly four of the four witnesses is a liar. Assume that each of the statements is either true or false.How many of the winesses are liars?

Determine all pairs of real numbers $(x,y)$ satisfying \begin{align*} x^3+9x^2y&=10,\\ y^3+xy^2 &=2. \end{align*}

A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?

Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$. Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Points $P$, $Q$, and $R$ are chosen on segments $BC$, $CA$, and $AB$, respectively, such that triangles $AQR$, $BPR$, $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$. What is the perimeter of $PQR$? [i]2021 CCA Math Bonanza Individual Round #2[/i]

Let $n \geq 2$ be an integer. Let $S$ be a subset of $\{1,2,\dots,n\}$ such that $S$ neither contains two elements one of which divides the other, nor contains two elements which are coprime. What is the maximal possible number of elements of such a set $S$?