We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$ Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\ (Ivan Novak)

Let $ABC$ be a triangle with $AB=72,AC=98,BC=110$, and circumcircle $\Gamma$, and let $M$ be the midpoint of arc $BC$ not containing $A$ on $\Gamma$. Let $A'$ be the reflection of $A$ over $BC$, and suppose $MB$ meets $AC$ at $D$, while $MC$ meets $AB$ at $E$. If $MA'$ meets $DE$ at $F$, find the distance from $F$ to the center of $\Gamma$. [i]Proposed by Michael Kural[/i]

If two faces of a dice have a common edge, the two faces are called adjacent faces. In how many ways can we construct a dice with six faces such that any two consecutive numbers lie on two adjacent faces? $\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \text{None}$

Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always 10 cm. Each side of the table is more than 1 meter long. At the initial moment both ants are on the same side of the table. (a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter? (b) [i](3 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ . Proposed by Alireza Dadgarnia

Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$. Find $\lim_{n\to\infty} f_n(x)$

The coordinates of $ A,B$ and $ C$ are $ (5,5),(2,1)$ and $ (0,k)$ respectively. The value of $ k$ that makes $ \overline{AC}\plus{}\overline{BC}$ as small as possible is: $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac{1}{2} \qquad\textbf{(C)}\ 3\frac{6}{7} \qquad\textbf{(D)}\ 4\frac{5}{6} \qquad\textbf{(E)}\ 2\frac{1}{7}$

Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have \[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \] Prove that \[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]

Let $a,b>1$ be odd positive integers. A board with $a$ rows and $b$ columns without fields $(2,1),(a-2,b)$ and $(a,b)$ is tiled with $2\times 2$ squares and $2\times 1$ dominoes (that can be rotated). Prove that the number of dominoes is at least $$\frac{3}{2}(a+b)-6.$$

For covering the floor of a rectangular room rectangular tiles of sizes $2 \times 2$ and $4 \times 1$ were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.

A permutation of $\{1, 2 \cdots, n \}$ is [i]magic[/i] if each element $k$ of it has at least $\left\lfloor \frac{k}{2} \right\rfloor$ numbers less to it at the left. For each $n$ find the number of [i]magical[/i] permutations.

Let $F_n$ be the $n$th Fibonacci number ($F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$). Construct infinitely many positive integers $n$ such that $n$ divides $F_{F_n}$ but $n$ does not divide $F_n$.

Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.

nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square. [i]Proposed by Mihir Singhal and Michael Kural[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $$f(xy+f(x))=f(xf(y))+x$$

How many structural isomers have the formula $\ce{C3H6Cl2}$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4$

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 15 $

Let $n>4$ be a positive integer, which is divisible by $4$. We denote by $A_n$ the sum of the odd positive divisors of $n$. We also denote $B_n$ the sum of the even positive divisors of $n$, excluding the number $n$ itself. Find the least possible value of the expression $$f(n)=B_n-2A_n,$$ for all possible values of $n$, as well as for which positive integers $n$ this minimum value is attained.

Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that $\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$; $\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$. Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.

Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$ Prove that $f$ is constant.

Let $ABCD$ be a non-isosceles trapezoid. Define a point $A1$ as intersection of circumcircle of triangle $BCD$ and line $AC$. (Choose $A_1$ distinct from $C$). Points $B_1, C_1, D_1$ are defined in similar way. Prove that $A_1B_1C_1D_1$ is a trapezoid as well.

Let $f(x)$ be a differentiable function defined on the interval $(0,1)$ such that $|f'(x)| \leq M$ for $0<x<1$ and a positive real number $M.$ Prove that $$\left| \int_{0}^{1} f(x)\; dx - \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n} \right) \right | \leq \frac{M}{n}.$$

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$