Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Let $(x_n)_{n\in Z}$ and $(y_n)_{n\in Z}$ be two sequences of integers such that $|x_{n+2} - x_n| \le 2$ and $x_n + x_m = y_{n^2+m^2}$ for all $n, m \in Z$. Show that the sequence of $x_n$s takes at most $6$ distinct values. (Paolo Leonetti)

Inside the parallelogram $ABCD$, point $M$ is chosen, and inside the triangle $AMD$, point $N$ is chosen in such a way that $$\angle MNA + \angle MCB =\angle MND + \angle MBC = 180^o.$$ Prove that lines $MN$ and $AB$ are parallel.

On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.

$ \frac{(.2)^{3}}{(.02)^{2}}= $ $ \text{(A)}\ .2\qquad\text{(B)}\ 2\qquad\text{(C)}\ 10\qquad\text{(D)}\ 15\qquad\text{(E)}\ 20 $

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

In trapezoid $ABCD$ , the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is [asy] draw((0,0)--(4,3)--(12,3)--(16,0)--cycle); draw((4,3)--(4,0),dashed); draw((3.2,0)--(3.2,.8)--(4,.8)); label("$A$",(0,0),SW); label("$B$",(4,3),NW); label("$C$",(12,3),NE); label("$D$",(16,0),SE); label("$8$",(8,3),N); label("$16$",(8,0),S); label("$3$",(4,1.5),E);[/asy] $ \text{(A)}\ 27\qquad\text{(B)}\ 30\qquad\text{(C)}\ 32\qquad\text{(D)}\ 34\qquad\text{(E)}\ 48 $

Let $U$ be the center of the circle inscribed in the triangle $ABC$, $O_1$, $O_2$ and $O_3$ the centers of the circles circumscribed by the triangles $BCU$, $CAU$ and $ABU$ respectively. Prove that the circles circumscribed around the triangles $ABC$ and $O_1O_2O_3$ have the same center.

Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,

Positive integers $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$ satisfy $2 \leq a_i \leq 166$ and $a_i^{b_i} \cong a_{i+1}^2$ (mod 167) for each $1 \leq i \leq 7$ (where $a_8=a_1$). Compute the minimum possible value of $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$.

Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$?

Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?

On a dance evening, each of the gentlemen present has sex with at least one of the ladies present danced and each of the ladies present danced with at least one of the gentlemen present. No gentleman has sex with every lady present and no lady has sex with every gentleman present danced. It must be proven that there were two such ladies and two such gentlemen among those present has that that evening each of the two ladies with exactly one of the two men, and each of the both men danced with exactly one of the two women. It is assumed that the dance evening did not take place without ladies and gentlemen, i.e. the crowd, which consists of all the ladies and gentlemen present, is not empty. [hide=original wording]An einem Tanzabend hat jeder der anwesenden Herren mit mindestens einer der anwesenden Damen getanzt und jede der anwesenden Damen mit mindestens einem der anwesenden Herren. Kein Herr hat mit jeder der anwesenden Damen und keine Dame mit jedem der anwesenden Herren getanzt. Es ist zu beweisen, dass es unter den Anwesenden zwei solche Damen und zwei solche Herren gegeben hat, dass an dem Abend jede der beiden Damen mit genau einem der beiden Herren, und jeder der beiden Herren mit genau einer der beiden Damen getanzt hat. Es wird vorausgesetzt, dass der Tanzabend nicht ohne Damen und Herren stattgefunden hat, d.h., die Menge, die aus allen anwesenden Damen und Herren besteht, ist nicht leer.[/hide]

Consider an acute triangle $ABC$. Let $D$, $E$ and $F$ be the feet of the altitudes through vertices $A$, $B$ and $C$. Denote by $A'$, $B'$, $C'$ the projections of $A$, $B$, $C$ onto lines $EF$, $FD$, $DE$, respectively. Further, let $H_D$, $H_E$, $H_F$ be the orthocenters of triangles $DB'C'$, $EC'A'$, $FA'B'$. Show that $$H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2.$$

Let $G$ be the intersection of medians of $\triangle ABC$ and $I$ be the incenter of $\triangle ABC$. If $|AB|=c$, $|AC|=b$ and $GI \perp BC$, what is $|BC|$? $ \textbf{(A)}\ \dfrac{b+c}2 \qquad\textbf{(B)}\ \dfrac{b+c}{3} \qquad\textbf{(C)}\ \dfrac{\sqrt{b^2+c^2}}{2} \qquad\textbf{(D)}\ \dfrac{\sqrt{b^2+c^2}}{3\sqrt 2} \qquad\textbf{(E)}\ \text{None of the preceding} $

For natural numbers $m,n$, set $a=(n+1)^m-n$ and $b=(n+1)^{m+3}-n$. (a) Prove that $a$ and $b$ are coprime if $m$ is not divisible by $3$. (b) Find all numbers $m,n$ for which $a$ and $b$ are not coprime.

Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

Let $a_1=3$, $a_2=7$, and $a_3=1$. Let $b_0=0$ and for all positive integers $n$, let $b_n=10b_{n-1}+a_n$. Compute $b_1\times b_2\times b_3$. [i]2020 CCA Math Bonanza Lightning Round #1.2[/i]

Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$, each $b_i$ equal $-1,0$ or $1$, but not all being $0$, such that the number $$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$ is divisible by $2015$.

For any positive integer $n$, we have the set $P_n = \{ n^k \mid k=0,1,2, \ldots \}$. For positive integers $a,b,c$, we define the group of $(a,b,c)$ as lucky if there is a positive integer $m$ such that $a-1$, $ab-12$, $abc-2015$ (the three numbers need not be different from each other) belong to the set $P_m$. Find the number of lucky groups.