For a triangle $ABC$ and a point $M$ inside the triangle we consider the lines $AM, BM,CM$ which intersect the sides $BC, CA, AB$ in $A_1, B_1, C_1$ respectively. Take $A', B', C'$ to be the intersection points between the lines $AA_1, BB_1, CC_1$ and $B_1C_1, C_1A_1, A_1B_1$ respectively. a) Prove that the lines $BC', CB'$ and $AA'$ intersect in a point $A_2$; b) Define similarly points $B_2, C_2$. Find the loci of $M$ such that the triangle $A_1B_1C_1$ is similar with the triangle $A_2B_2C_2$.

Let $\omega$ be the circumcircle of an acute angled tirangle $ABC.$ The line tangent to $\omega$ at $A$ intersects the line $BC$ at the point $T.$ Let the midpoint of segment $AT$ be $N,$ and the centroid of $\triangle ABC$ be the point $G.$ The other tangent line drawn from $N$ to $\omega$ intersects $\omega$ at the point $L.$ The line $LG$ meets $\omega$ at $S\neq L.$ Prove that $AS\parallel BC.$

Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$. Show that $O$ lies on the line through the midpoints of $AC$ and $BD$.

Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.

Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $ [i]Ovidiu Pop[/i]

In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$? [asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$X$", X, dir(180)); label("$W$", W, dir(180)); label("$Y$", Y, dir(0)); label("$Z$", Z, dir(0)); draw(W--X--Y--Z--cycle); dot(X); dot(Y); dot(W); dot(Z); pair M = (2, 0); pair A = (8, 3); label("$A$", A, dir(0)); dot(M); dot(A); draw(W--M--A--cycle); markscalefactor = 0.05; draw(rightanglemark(W, M, A)); label("$M$", M, dir(-90)); [/asy] $ \textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad $

In a chess tournament , every two participants play each other exactly once. A win is worth one point , a draw is worth half a point and a loss is worth zero points. Looking back at the end of the tournament, a game is called an upset if the total number of points obtained by the winner of that game is less than the total number of points obtained by the loser of that game. (a) Prove that the number of upsets is always strictly less than three-quarters of the total number of games in the tournament. (b) Prove that three-quarters cannot be replaced by a smaller number. (S Tokarev) PS. part (a) for Juniors, both parts for Seniors

Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$

$ABCD$ is a square side $10$. There are four points $P_1, P_2, P_3, P_4$ inside the square. Show that we can always construct line segments parallel to the sides of the square of total length $25$ or less, so that each $P_i$ is linked by the segments to both of the sides $AB$ and $CD$. Show that for some points $P_i$ it is not possible with a total length less than $25$.

In a triangle $ABC,$ point $D$ lies on $AB$. It is given that $AD=25, BD=24, BC=28, CD=20. AC=?$

In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?

In the parallelolgram A$BCD$ with the center $S$, let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$ be the touch point with the diagonal $BD$. Prove that the lines $OS$ and $CT$ are parallel.

The hypotenuse $ c$ and one arm $ a$ of a right triangle are consecutive integers. The square of the second arm is: $ \textbf{(A)}\ ca \qquad\textbf{(B)}\ \frac {c}{a} \qquad\textbf{(C)}\ c \plus{} a \qquad\textbf{(D)}\ c \minus{} a \qquad\textbf{(E)}\ \text{none of these}$

Let $N_6$ be the answer to problem 6. Given positive integers $n$ and $a$, the $n$[i]th tetration of[/i] $a$ is defined as \[ ^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}. \] For example, $^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536$. Compute the units digit of $^{2024}N_6$.

The sum of two natural numbers is $17,402.$ One of the two numbers is divisible by $10.$ If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? $\textbf{(A) }10,272 \qquad \textbf{(B) }11,700 \qquad \textbf{(C) }13,362 \qquad \textbf{(D) }14,238 \qquad \textbf{(E) }15,426$

Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system \begin{align*} x_1x_2 &= a_1, \\ x_2x_3 &= a_2, \\ &\ \vdots \\ x_{n-1}x_n &= a_{n-1}, \\ x_nx_1 &\ge a_n. \end{align*}

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

Let $ P$ be a regular polygon. A regular sub-polygon of $ P$ is a subset of vertices of $ P$ with at least two vertices such that divides the circumcircle to equal arcs. Prove that there is a subset of vertices of $ P$ such that its intersection with each regular sub-polygon has even number of vertices.

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee[/i]

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

Seven congruent line segments are connected together at their endpoints as shown in the figure below at the left. By raising point $E$ the linkage can be made taller, as shown in the figure below and to the right. Continuing to raise $E$ in this manner, it is possible to use the linkage to make $A, C, F$, and $E$ collinear, while simultaneously making $B, G, D$, and $E$ collinear, thereby constructing a new triangle $ABE$. Prove that a regular polygon with center $E$ can be formed from a number of copies of this new triangle $ABE$, joined together at point $E$, and without overlapping interiors. Also find the number of sides of this polygon and justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/2/6/b3826b7ba7ea49642477878a03ac590281df43.png[/img]

How many distinct prime factors does the number $36$ have? $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }9\qquad\textbf{(E) }15$

A four-digit positive integer is called [i]doubly[/i] if its first two digits form some permutation of its last two digits. For example, $1331$ and $2121$ are both [i]doubly[/i]. How many four-digit [i]doubly[/i] positive integers are there?

The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation: \[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \] for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.

In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $2$? [i]G.M.Sharafetdinova[/i]