A natural number $n$ is given and written in a row of $n$ numbers, each of which is equal to $0$ or $1$. Then $n - 1$ numbers are written below in a row - one number under each pair of adjacent numbers of the first row. At the same time, $0$ is written under a pair of identical numbers. and under a pair of different ones $1$. Then, under the second row, the third of $n- 2$ numbers is similarly written, etc., until we get a triangular table with $n$ rows. For a given $n$, find the largest possible number of units in such a table.

Flavian and Pavel play a game. Starting with Flavian, they take turns eliminating exactly one edge from a complete graph with $2024$ vertices. The first player to make a move that leaves no cycles loses. Determine who has a winning strategy. [i]Proposed by Pavel Ciurea[/i]

Hamza and Majid play a game on a horizontal $3 \times 2015$ white board. They alternate turns, with Hamza going first. A legal move for Hamza consists of painting three unit squares forming a horizontal $1 \times 3$ rectangle. A legal move for Majid consists of painting three unit squares forming a vertical $3\times 1$ rectangle. No one of the two players is allowed to repaint already painted squares. The last player to make a legal move wins. Which of the two players, Hamza or Majid, can guarantee a win no matter what strategy his opponent chooses and what is his strategy to guarantee a win? Lê Anh Vinh

Given three letters $X, Y, Z$, we can construct letter sequences arbitrarily, such as $XZ, ZZYXYY, XXYZX$, etc. For any given sequence, we can perform following operations: $T_1$: If the right-most letter is $Y$, then we can add $YZ$ after it, for example, $T_1(XYZXXY) = (XYZXXYYZ).$ $T_2$: If The sequence contains $YYY$, we can replace them by $Z$, for example, $T_2(XXYYZYYYX) = (XXYYZZX).$ $T_3$: We can replace $Xp$ ($p$ is any sub-sequence) by $XpX$, for example, $T_3(XXYZ) = (XXYZX).$ $T_4$: In a sequence containing one or more $Z$, we can replace the first $Z$ by $XY$, for example, $T_4(XXYYZZX) = (XXYYXYZX).$ $T_5$: We can replace any of $XX, YY, ZZ$ by $X$, for example, $T_5(ZZYXYY) = (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$ Using above operations, can we get $XYZZ$ from $XYZ \ ?$

$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? [img]https://s8.uupload.ir/files/photo14908575660_2r3g.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that: $$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$ For what kind of triangles does the equality hold?

Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic. [I]Proposed by S. Berlov[/i]

Given an acute angle and a circle inside the angle. Find a point $ M $ on the circle such that the sum of the distances of the point $ M $ from the sides of the angle is a minimum.

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

In a triangle $ABC$, the points $A’$, $B’$ and $C’$ on the sides opposite $A$ ,$B$ and $C$, respectively, are such that the lines $AA’$, $BB’$ and $CC’$ are concurrent. Prove that the diameter of the circumscribed circle of the triangle $ABC$ equals the product $|AB’|\cdot |BC’|\cdot |CA’|$ divided by the area of the triangle $A’B’C’$.

Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples. [i]Mitchell Lee.[/i]

Let $A = {1,2,3,4,5}$ and $B = {0,1,2}$. Find the number of pairs of functions ${{f,g}}$ where both f and g map the set A into the set B and there are exactly two elements $x \in A$ where $f(x) = g(x)$. For example, the function f that maps $1 \rightarrow 0,2 \rightarrow 1,3 \rightarrow 0,4 \rightarrow 2,5 \rightarrow 1$ and the constant function g which maps each element of A to 0 form such a pair of functions.

Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$ [img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]

Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number.

Let $PQRS$ be a square with side length 12. Point $A$ lies on segment $\overline{QR}$ with $\angle QPA = 30^\circ$, and point $B$ lies on segment $\overline{PQ}$ with $\angle SRB = 60^\circ$. What is $AB$?

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

Let $P$ be the intersection point of the diagonals of the quadrangle $ABCD$, $M$ the intersection point of the lines connecting the midpoints of its opposite sides, $O$ the intersection point of the perpendicular bisectors of the diagonals, $H$ the intersection point of the lines connecting the orthocenters of the triangles $APD$ and $BCP$, $APB$ and $CPD$. Prove that $M$ is the midpoint of $OH$.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Let $a,b$ and $c$ be the sides of a right angled triangle. Let $\theta$ be the smallest angle of this triangle. If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are also the sides of a right angled triangle then show that $\sin\theta=\frac{\sqrt{5}-1}{2}$

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 01[/b] Find all $(x,y,z)$ positive integers, such that: $\sqrt{\frac{2006}{x+y}} + \sqrt{\frac{2006}{y+z}} + \sqrt{\frac{2006}{z+x}},$ is an integer. --- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88509]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

If a number $N$, $N\neq 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is: $\textbf{(A) }\dfrac1R\qquad \textbf{(B) }R\qquad \textbf{(C) }4\qquad \textbf{(D) }\dfrac14\qquad \textbf{(E) }-R$

Find the parallel displacement of a vector pointing north at Leningrad (latitude $60^{\circ}$) from west to east along a closed parallel.

A pirate has five purses with 30 coins in each. He knows that one purse contains only gold coins, another one contains only silver coins, the third one contains only bronze coins, and the remaining two ones contain 10 gold, 10 silver and 10 bronze coins each. It is allowed to simultaneously take one or several coins out of any purses (only once), and examine them. What is the minimal number of taken coins that is necessary to determine for sure the content of at least one purse? [i]Mikhail Evdokimov[/i]

Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form $2x^{2}-3y^{2}$ with integral $x$ and $y$ or that of positive integers not exceeding 1000000 that can be represented by the form $10xy-x^{2}-y^{2}$ with integral $x$ and $y?$ [i]Proposed by A. Golovanov[/i]