Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?

find the smallest integer $n\geq1$ such that the equation : $$a^2+b^2+c^2-nd^2=0 $$ has $(0,0,0,0)$ as unique solution .

Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$.

a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$ We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$ Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$ b) Prove that the product of $2$ original polynomials is a original polynomial.

Consider these two geoboard quadrilaterals. Which of the following statements is true? [asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 5; ++b) { dot((a,b)); } } draw((0,3)--(0,4)--(1,3)--(1,2)--cycle); draw((2,1)--(4,2)--(3,1)--(3,0)--cycle); label("I",(0.4,3),E); label("II",(2.9,1),W); [/asy] $\text{(A)}\ \text{The area of quadrilateral I is more than the area of quadrilateral II.}$ $\text{(B)}\ \text{The area of quadrilateral I is less than the area of quadrilateral II.}$ $\text{(C)}\ \text{The quadrilaterals have the same area and the same perimeter.}$ $\text{(D)}\ \text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$ $\text{(E)}\ \text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$

You have the following pieces: one $4\times 1$ rectangle, two $3\times 1$ rectangles, three $2\times 1$ rectangles, and four $1\times 1$ squares. Ariel and Bernardo play the following game on a board of $n\times n$, where $n$ is a number that Ariel chooses. In each move, Bernardo receives a piece $R$ from Ariel. Next, Bernardo analyzes if he can place $R$ on the board so that it has no points in common with any of the previously placed pieces (not even a common vertex). If there is such a location for $R$, Bernardo must choose one of them and place $R$. The game stops if it is impossible to place $R$ in the way explained, and Bernardo wins. Ariel wins only if all $10$ pieces have been placed on the board. a) Suppose Ariel gives Bernardo the pieces in decreasing order of size. What is the smallest n that guarantees Ariel victory? b) For the $n$ found in a), if Bernardo receives the pieces in increasing order of size, is Ariel guaranteed victory? Note: Each piece must cover exactly a number of unit squares on the board equal to its own size. The sides of the pieces can coincide with parts of the edge of the board.

The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].

Cheryl rolls a fair dice twice. If the dice's six faces are numbered by $1$, $2$, $3$, $4$, $5$, $6$, what is the probability that the number on one of her rolls is a divisor of the number on the other roll? $\textbf{(A) }\dfrac29\qquad\textbf{(B) }\dfrac5{18}\qquad\textbf{(C) }\dfrac49\qquad\textbf{(D) }\dfrac12\qquad\textbf{(E) }\dfrac{11}{18}$

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$? $\textbf{(A) }10\qquad\textbf{(B) }18\qquad\textbf{(C) }25\qquad\textbf{(D) }36\qquad\textbf{(E) }81$

The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .

For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

The inscribed circle $\omega$ of triangle $ABC$ with center $I$ touches the sides $AB, BC, CA$ at points $C_1, A_1, B_1$. The circle circumsrcibed around $\vartriangle AB_1C_1$ intersects the circumscribed circle of $ABC$ for second time at the point $K$. Let $M$ be the midpoint $BC$, $L$ be the midpoint of $B_1C_1$. The circle circumsrcibed around $\vartriangle KA_1M$ cuts intersects $\omega$ for second time at the point $T$. Prove that the circumscribed circles of triangles $KLT$ and $LIM$ are tangent.

A jeweler can get an alloy that is $40\%$ gold for $200$ dollars per ounce, an alloy that is $60\%$ gold for $300$ dollar per ounce, and an alloy that is $90\%$ gold for $400$ dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is $50\%$ gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.

$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.

The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$. I. Voronovich

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

A non-isosceles triangle $\triangle ABC$ has incenter $I$ and the incircle hits $BC, CA, AB$ at $D, E, F$. Let $EF$ hit the circumcircle of $CEI$ at $P \not= E$. Prove that $\triangle ABC = 2 \triangle ABP$.

At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle. (A. Zaslavsky)

Let $ D,E,F$ be tangency point of incircle of triangle $ ABC$ with sides $ BC,AC,AB$. $ DE$ and $ DF$ intersect the line from $ A$ parallel to $ BC$ at $ K$ and $ L$. Prove that the Euler line of triangle $ DKL$ passes through Feuerbach point of triangle $ ABC$.

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.