Determine all functions $ f$ defined on the set of rational numbers that take rational values for which \[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y, \] for each $ x$ and $ y$.

Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$ .

Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$? [i] (Adapted from Archimedes 2011, Traian Preda)[/i]

Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between $1$ and $5$ letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word! What is the maximum number of possible words in Runese?

Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$, and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.

Find positive integers $n$ and $a_1, a_2, \dots, a_n$ such that $$a_1+a_2+\dots a_n=1979$$ and the product $a_1a_2\dots a_n$ as large as possible.

Determine all positive real solutions of the following system of equations: $$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$ $$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$

Let $ABCD$ be a quadrilateral $\angle B = \angle D$ and $AD = CD$. The incircle of triangle $ABC$ touches the sides $BC$ and $AB$ at points $E$ and $F$ respectively. Prove that the midpoints of segments $AC, BD, AE,$ and $CF$ are concyclic.

Let $ABC$ be an acute-angled triangle and $D$ a point on the altitude through $C$. Let $E$, $F$, $G$ and $H$ be the midpoints of the segments $AD$, $BD$, $BC$ and $AC$. Show that $E$, $F$, $G$, and $H$ form a rectangle. (G. Anegg, Innsbruck)

In triangle $ABC$ $CL$ is a bisector($L$ lies $AB$) $I$ is center incircle of $ABC$. $G$ is intersection medians. If $a=BC, b=AC, c=AB$ and $CL\perp GI$ then prove that $\frac{a+b+c}{3}=\frac{2ab}{a+b}$

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

The equation $ x^3\plus{}6x^2\plus{}11x\plus{}6\equal{}0$ has: $ \textbf{(A)}\ \text{no negative real roots} \qquad \textbf{(B)}\ \text{no positive real roots} \qquad \textbf{(C)}\ \text{no real roots} \\ \textbf{(D)}\ \text{1 positive and 2 negative roots} \qquad \textbf{(E)}\ \text{2 positive and 1 negative root}$

Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$. Define $P_B$ and $P_C$ similarly. Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.

Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$

In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

Two children take turns breaking chocolate bar that is 5*10 squares. They can only break the bar using the divisions between squares and can only do 1 break at a time.. The first player that when breaking the chocolate bar breaks off only a single square wins. Is there a winning strategy for any player?

The incircle of $ \triangle ABC$ touches $ BC$, $ AC$, and $ AB$ at $ A_1$, $ B_1$, and $ C_1$, respectively. The line $ AA_1$ intersects the incircle at $ Q$, again. $ A_1C_1$ and $ A_1B_1$ intersect the line, passing through $ A$ and parallel to $ BC$, at $ P$ and $ R$, respectively. If $ \angle PQC_1 \equal{} 45^\circ$ and $ \angle RQB_1 \equal{} 65^\circ$, then $ \angle PQR$ will be ? $\textbf{(A)}\ 110^\circ \qquad\textbf{(B)}\ 115^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 125^\circ \qquad\textbf{(E)}\ 130^\circ$

Let $ABC$ be a triangle. Suppose that the perpendicular bisector of $BC$ meets the circle of diameter $AB$ at a point $D$ at the opposite side of $BC$ with respect to $A$, and meets the circle through $A, C, D$ again at $E$. Prove that $\angle ACE=\angle BCD$. [i]Proposed by José Manuel Guerra and Victor Domínguez[/i]

Suppose $a,b,c>0$ and $abc=64$, show that $$\sum_{cyc}\frac{a^2}{\sqrt{a^3+8}\sqrt{b^3+8}}\ge\frac{2}{3}$$

In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.

A segment $AB$ of length $100$ is divided into $100$ little segments of length $1$ by $99$ intermediate points. Endpoint $A$ is assigned $0$ and endpoint $B$ is assigned $1$. Gustavo assigns each of the $99$ intermediate points a $0$ or a $1$, at his choice, and then color each segment of length $1$ blue or red, respecting the following rule: [i]The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. [/i] Determine if Gustavo can assign the $0$'s and $1$'s so as to get exactly $30$ blue segments. And $35$ blue segments? (In each case, if the answer is yes, show a distribution of $0$'s and $1$'s, and if the answer is no, explain why).

Let $\Omega$ be a family of subsets of $M$ such that: $(\text{i})$ If $|A\cap B|\ge 2$ for $A,B\in\Omega$, then $A=B$; $(\text{ii})$ There exist different subsets $A,B,C\in\Omega$ with $|A\cap B\cap C|=1$; $(\text{iii})$ For every $A\in\Omega$ and $a\in M \ A$, there is a unique $B\in\Omega$ such that $a\in B$ and $A\cap B=\emptyset$. Prove that there are numbers $p$ and $s$ such that: $(1)$ Each $a\in M$ is contained in exactly $p$ sets in $\Omega$; $(2)$ $|A|=s$ for all $A\in\Omega$; $(3)$ $s+1\ge p$.

A right circular cone has base radius 1 cm and slant height 3 cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back to $P$ is drawn (see diagram). What is the minimum distance from the vertex $V$ to this path? [asy] import graph; unitsize(1 cm); filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen); draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed); draw(yscale(0.3)*(arc((0,0),1.5,180,360))); draw((1.5,0)--(0,4)--(-1.5,0)); draw((0,0)--(1.5,0),Arrows); draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows); draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed); draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360))); label("$V$", (0,4), N); label("1 cm", (0.75,-0.5), N); label("$P$", (-1.5,0), SW); label("3 cm", (1.7,2)); [/asy]

Let $ABCD$ be a rectangle, $AB = a, BC = b$. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being $d$) that are at an the angle $\phi, 0 \leq \phi \leq 90^{\circ},$ with respect to $AB$. Let $L$ be the sum of the lengths of all the segments intersecting the rectangle. Find: [i](a)[/i] how $L $ varies, [i](b)[/i] a necessary and sufficient condition for $L$ to be a constant, and [i](c)[/i] the value of this constant.

Prove that the number $\binom np-\left\lfloor\frac np\right\rfloor$ is divisible by $p$ for every prime number and integer $n\ge p$.