Let there be a figure with $9$ disks and $11$ edges, as shown below. We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.

In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$? $ \begin{array}{ccccccc} A&B&C&D&E&D&B\\ &B&C&D&E&D&B\\ &&C&D&E&D&B\\ &&&D&E&D&B\\ &&&&E&D&B\\ &&&&&D&B\\ +&&&&&&B\\ \hline A&A&A&A&A&A&A \end{array} $

A palindrome in base $b$ is a positive integer whose base-$b$ digits read the same backwards and forwards; for example, $2002$ is a $4$-digit palindrome in base $10$. Note that $200$ is not a palindrome in base $10$, but it is a $3$-digit palindrome: $242$ in base $9$, and $404$ in base $7$. Prove that there is an integer which is a $3$-digit palindrome in base $b$ for at least $2002$ different values of $b$.

Let $k$ be a positive integer, and let $a,b$ and $c$ be positive real numbers. Show that \[a(1-a^k)+b(1-(a+b)^k)+c(1-(a+b+c)^k)<\frac{k}{k+1}.\] [i]* * *[/i]

Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$

$(BEL 3)$ Construct the circle that is tangent to three given circles.

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

Does there exist an angle $ \alpha\in(0,\pi/2)$ such that $ \sin\alpha$, $ \cos\alpha$, $ \tan\alpha$ and $ \cot\alpha$, taken in some order, are consecutive terms of an arithmetic progression?

Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$. Lê Phúc Lữ

Meghal is playing a game with $2016$ rounds $1, 2, ..., 201$6. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2\pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?

Find all triplets $(a, b, c)$ of positive integers, such that $a+bc, b+ac, c+ab$ are primes and all divide $(a^2+1)(b^2+1)(c^2+1)$.

Two circles $\omega_1$ and $\omega_2$ are externally tangent at a point $A$. Let $\ell$ be a line tangent to $\omega_1$ at $B\neq A$ and $\omega_2$ at $C\neq A$. Let $BX$ and $CY$ be diameters in $\omega_1$ and $\omega_2$ respectively. Suppose points $P$ and $Q$ lies on $\omega_2$ such that $XP$ and $XQ$ are tangent to $\omega_2$, and points $R$ and $S$ lies on $\omega_1$ such that $YR$ and $YS$ are tangent to $\omega_1$. a) Prove that the points $P$, $Q$, $R$, $S$ lie on a circle $\Gamma$. b) Prove that the four segments $XP$, $XQ$, $YR$, $YS$ determine a quadrilateral with an incircle $\gamma$, and its radius is $\displaystyle\frac{1}{\sqrt{5}}$ times the radius of $\Gamma$. [i]Proposed by Ivan Chan Kai Chin[/i]

Rational numbers $a$ and $b$ satisfy the equality $$a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0. $$ Prove that the number $1-ab$ is the square of the rational numbers.

For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.

In $ \triangle ABC$ , $ AB \equal{} 13$, $ AC \equal{} 5$, and $ BC \equal{} 12$. Points $ M$ and $ N$ lie on $ \overline{AC}$ and $ \overline{BC}$, respectively, with $ CM \equal{} CN \equal{} 4$. Points $ J$ and $ K$ are on $ \overline{AB}$ so that $ \overline{MJ}$ and $ \overline{NK}$ are perpendicular to $ \overline{AB}$. What is the area of pentagon $ CMJKN$? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair C=(0,0), B=(12,0), A=(0,5), M=(0,4), Np=(4,0); pair K=foot(Np,A,B), J=foot(M,A,B); draw(A--B--C--cycle); draw(M--J); draw(Np--K); label("$C$",C,SW); label("$A$",A,NW); label("$B$",B,SE); label("$N$",Np,S); label("$M$",M,W); label("$J$",J,NE); label("$K$",K,NE);[/asy]$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ \frac{81}{5} \qquad \textbf{(C)}\ \frac{205}{12} \qquad \textbf{(D)}\ \frac{240}{13} \qquad \textbf{(E)}\ 20$

Prove that there is no polyhedron whose every plane section is a triangle.