Let $\triangle ABC$ have incenter $I$ and centroid $G$. Suppose that $P_A$ is the foot of the perpendicular from $C$ to the exterior angle bisector of $B$, and $Q_A$ is the foot of the perpendicular from $B$ to the exterior angle bisector of $C$. Define $P_B$, $P_C$, $Q_B$, and $Q_C$ similarly. Show that $P_A, P_B, P_C, Q_A, Q_B,$ and $Q_C$ lie on a circle whose center is on line $IG$.

What is the smallest positive integer with exactly $7$ distinct proper divisors?

In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a\equal{}90^\circ\plus{}\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL\equal{}2R$. ($ R$ is the circumradius of $ ABC$)

Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent . a) Prove that there are consistent sequences other than $a_n = n$. b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ? c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?

Two of the numbers $a+b, a-b, ab, a/b$ are positive, the other two are negative. Find the sign of $b$

Let $n, \in \mathbb {N^*} , n\ge 3$ a) Prove that the polynomial $f (x)=\frac {X^{2^n-1}-1}{X-1}-X^n $ has a divisor of form $X^p +1$ where $p\in\mathbb {N^*} $ b) Show that for $n=7$ the polynomial $f (X) $ has three divisors with integer coefficients .

let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are the midpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQ through T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of the incircle . prove IK is parallel to AL

For real numbers $ x_1, x_2, \dots, x_n $ and $ y_1, y_2, \dots, y_n $ , the inequalities hold $ x_1 \geq x_2 \geq \ldots \geq x_n> 0 $ and $$ y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n. $$ Prove that $ ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n $.

Show that a triangle whose side lengths are prime numbers cannot have integer area.

Determine the last four digits of a perfect square of a natural number, knowing that the last three of them are the same.

An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will [i]not[/i] have distinct positive real roots? $\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$

Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.

Solve the system of equation $$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$

Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that \[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\] but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.

Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=?

There are twenty-four 4-digit numbers that use each of the four digits 2, 5, 7, and 4exactly once. Listed in numerical order from smallest to largest, the number in the $17th$ position in the list is $ \text{(A)}\ 4527\qquad\text{(B)}\ 5724\qquad\text{(C)}\ 5742\qquad\text{(D)}\ 7245\qquad\text{(E)}\ 7524 $

Inside a rhombus $ABCD$ with $\angle BAD=60$, points $F,H,G$ are choosen on lines $AD,DC,AC$ respectivily such that $DFGH$ is a paralelogram. Show that $BFH$ is a equilateral triangle.

Let ${\mathcal A}$ be a ${C^{\ast}}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$. Consider the operation \[ \displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+\] Prove that if for all ${x,y \in \mathcal A_+}$ we have \[ \displaystyle (x\circ y)\circ y = x \circ (y \circ y), \] then ${\mathcal A}$ is commutative. [i]Proposed by Lajos Molnár[/i]

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer and no points are awarded or deducted for each unanswered question. Joana obtained $87$ points. How many questions did she not answer?

In a group of nine persons it is not possible to choose four persons such that every one knows the three others. Prove that this group of nine persons can be partitioned into four groups such that nobody knows anyone from his or her group.

Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly. Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.

Right triangle $\triangle{DEF}$ with $\angle{D}=90^\circ$ and $\angle{F}=30^\circ$ is inscribed in equilateral triangle $\triangle{ABC}$ such that $D, E,$ and $F$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. Given that $BD=7$ and $DC=4,$ compute $DE.$