Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

Find all positive integers $a,b,c$ and prime $p$ satisfying that \[ 2^a p^b=(p+2)^c+1.\]

In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$. Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$. The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$. Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

p1. A quadratic equation has the natural roots $a$ and $ b$. Another quadratic equation has roots $ b$ and $c$ with $a\ne c$. If $a$, $ b$, and $c$ are prime numbers less than $15$, how many triplets $(a,b,c)$ that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to $ 1$)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The [i]Archipelago Fraction[/i] is a fraction $\frac{a}{b}$ such that $a$ and $ b$ are natural numbers with $a < b$. Find the sum of all Archipelago Fractions starting from a fraction with $b = 2$ to $b = 1000$. p3. Look at the following picture. The letters $a, b, c, d$, and $e$ in the box will replaced with numbers from $1, 2, 3, 4, 5, 6, 7, 8$, or $9$, provided that $a,b, c, d$, and $e$ must be different. If it is known that $ae = bd$, how many arrangements are there? [img]https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png[/img] p4. Given a triangle $ABC$ with $A$ as the vertex and $BC$ as the base. Point $P$ lies on the side $CA$. From point $A$ a line parallel to $PB$ is drawn and intersects extension of the base at point $D$. Point $E$ lies on the base so that $CE : ED = 2 :3$. If $F$ is the midpoint between $E$ and $C$, and the area of ​​triangle ABC is equal with $35$ cm$^2$, what is the area of ​​triangle $PEF$? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to $1001$, find the sum of all the numbers written on the sides of the cube.

Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is $ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$

Given an isosceles trapezoid and draw one of its diagonals (so we get $2$ triangles). On each triangle we get, there is an ant walking along the edge. There speed are equal and unchanged. These $2$ ants also didn't change their walking directions and their directions are opposite. Prove that: for all initial locations of the ants, they will meet each other at some time.

Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.

If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$?

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that $$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$ (b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)

Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

Find the number of trapeziums that it can be formed with the vertices of a regular polygon.

A round robin tournament is held with $2016$ participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of $k$ rounds. Find the minimum possible value of $k$ for which this can always be guaranteed. [i]Proposed by Nathan Ramesh

Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?

Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$

In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$

The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.

$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.

Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.

Consider the following expression $$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$ Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).