Let $ABC$ and $XYZ$ be two triangles. Define \[A_1=BC\cap ZX, A_2=BC\cap XY,\]\[B_1=CA\cap XY, B_2=CA\cap YZ,\]\[C_1=AB\cap YZ, C_2=AB\cap ZX.\] Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$. Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

Given is a cyclic pentagon $ABCDE$, inscribed in a circle $k$. The line $CD$ intersects $AB$ and $AE$ in $X$ and $Y$ respectively. Segments $EX$ and $BY$ intersect again at $P$, and they intersect $k$ in $Q$ and $R$, respectively. Point $A'$ is reflection of $A$ across $CD$. The circles $(PQR)$ and $(A'XY)$ intersect at $M$ and $N$. Prove that $CM$ and $DN$ intersect on $(PQR)$.

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Are there numbers $a, b$ such that $| a -b |\le 1979$ and the equation $ax^2 + (a + b) x + b = x$ has no roots?

[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $ [b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $

Let $a, b$ be positive real numbers satisfying $a + b = 1$. Show that if $x_1,x_2,...,x_5$ are positive real numbers such that $x_1x_2...x_5 = 1$, then $(ax_1+b)(ax_2+b)...(ax_5+b)>1$

Let $a$, $b$, $c$ be real numbers. Prove that \[ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.\]

Let $f(x) = x-\tfrac1{x}$, and define $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.

The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? $ \textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\qquad\textbf{(E)}\ 192 $

Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $x_0 = (0, 0)$, and for every $n \ge 1$, his jump is $x_n - x_{n - 1} = (\pm f(n), \pm g(n))$ or $(\pm g(n), \pm f(n)),$ with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0, 0)$ infinitely many times when (a) $f, g$ are polynomials with integer coefficients? (b) $f, g$ are any pair of functions from the positive integers to the integers?

Solve in real numbers the equation \[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]

In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively,such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$ and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.

Below is a net consisting of $3$ squares, $4$ equilateral triangles, and $1$ regular hexagon. Each polygon has side length $1$. When we fold this net to form a polyhedron, what is the volume of the polyhedron? (This figure is called a "triangular cupola".) Net: [asy] pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P; A = origin; B = (0,3); C = 3*dir(150); D = (0,1); E = (0,2); F = C+2*dir(30); G = C+dir(30); H = 2*dir(150); I = dir(150); J = (1,1); K = J+dir(30); L = (1,2); M = F+dir(120); N = G+dir(120); O = H+dir(240); P = I+dir(240); draw(A--B--C--cycle); draw(D--E--F--G--H--I--cycle); draw(D--E--L--J--cycle); draw(F--G--N--M--cycle); draw(H--I--P--O--cycle); draw(J--K--L--cycle); [/asy] Resulting polyhedron: [img]https://upload.wikimedia.org/wikipedia/commons/9/93/Triangular_cupola.png[/img]

The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is $\mu$. Which of the following gives the maximum value of $\theta$ such that the sign will not collapse? $\textbf{(A) } \sin \theta = 2 \mu \\ \textbf{(B) } \sin \theta /2 = \mu / 2\\ \textbf{(C) } \tan \theta / 2 = \mu\\ \textbf{(D) } \tan \theta = 2 \mu \\ \textbf{(E) } \tan \theta / 2 = 2 \mu$

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

In any $\vartriangle ABC, \ell$ is any line through $C$ and points $P, Q$. If $BP, AQ$ are perpendicular to the line $\ell$ and $M$ is the midpoint of the line segment $AB$, then prove that $MP = MQ$

Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that $$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$ Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.

Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.

Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

Alice needs to replace a light bulb located $10$ centimeters below the ceiling of her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters? $ \textbf{(A)}\ 32 \qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 40 $

Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number.

Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$

Let $ABCD$ an inscribed quadrilateral and $r$ and $s$ the reflections of the straight line through $A$ and $B$ over the inner angle bisectors of angles $\angle{CAD}$ and $\angle{CBD}$, respectively. Let $P$ the point of intersection of $r$ and $s$ and let $O$ the circumcentre of $ABCD$. Prove that $OP \perp CD$.