Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular. [i]Alternative formulation.[/i] The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.

Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$. [i]Proposed by Michael Kural[/i]

Let $a,b,c$ be some complex numbers. Prove that $$|\dfrac{a^2}{ab+ac-bc}| + |\dfrac{b^2}{ba+bc-ac}| + |\dfrac{c^2}{ca+cb-ab}| \ge \dfrac{3}{2}$$ if the denominators are not 0

Without using tables, find the exact value of the product: \[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]

Prove that $$\operatorname{Re}\left(\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)+\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)=\frac{7\pi^2}{72}-\frac{\ln^23}8$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ [i]Proposed by Paolo Perfetti[/i]

Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.

If $a$ and $b$ are complex numbers such that $a^2 + b^2 = 5$ and $a^3 + b^3 = 7$, then their sum, $a + b$, is real. The greatest possible value for the sum $a + b$ is $\tfrac{m+\sqrt{n}}{2}$ where $m$ and $n$ are integers. Find $n.$

Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.

Given $a$, $b$, and $c$ are complex numbers satisfying \[ a^2+ab+b^2=1+i \] \[ b^2+bc+c^2=-2 \] \[ c^2+ca+a^2=1, \] compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)

Find the value of $2(\sin2^{\circ}\tan1^{\circ}+\sin4^{\circ}\tan1^{\circ}+\cdots+\sin178^{\circ}\tan 1^{\circ})$

A positive integer $n\geq 4$ is called [i]interesting[/i] if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$

Find the area of the set of all points $ z $ in the complex plane that satisfy $ \left| z - 3i \right| + \left| z - 4 \right| \leq 5\sqrt{2} $.

Find all complex numbers $m$ such that polynomial \[x^3 + y^3 + z^3 + mxyz\] can be represented as the product of three linear trinomials.

Let $ABC$ be a triangle. Let $A_1$ and $A_2$ be points on side $BC, B_1$ and $B_2$ be points on side $CA$ and $C_1$ and $C_2$ be points on side $AB$ such that $A_1A_2B_1B_2C_1C_2$ is a convex hexagon and that $B,A_1,A_2$ and $C$ are located in that order on side $BC$. We say that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if there exists a triangle $PQR$ and there exist $X,Y$ and $Z$ on sides $QR, RP$ and $PQ$ respectively, such that triangle $AB_2C_1$ is congruent in that order to triangle $PYZ$, triangle $BA_1C_2$ is congruent in that order to triangle $QXZ$ and triangle $CA_2B_1$ is congruent in that order to triangle $RXY$. Prove that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if and only if the centroids of triangles $A_1B_1C_1$ and $A_2B_2C_2$ coincide.

If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a $ \textbf{(A)}\text{ right triangle}\qquad\textbf{(B)}\text{ circle}\qquad\textbf{(C)}\text{ hyperbola}\qquad\textbf{(D)}\text{ line}\qquad\textbf{(E)}\text{ parabola} $

Let $z$ be a complex number such that $z,z^2,z^3$ are all collinear in the complex plane . Show that $z$ is a real number .

Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$.

For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$.

Solve the equation $ \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2$.

Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon. [i]Daniel Jinga[/i]

[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $ [b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that $$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$ for any nonnegative integer $ n. $

Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that \[ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. \]