The set $\{1, 2, 3, 4\}$ can be partitioned into two subsets $A = \{1, 4\}$ and $B = \{3, 2\}$ with no common elements and such that the sum of the elements of $A$ is equal to the sum of the elements of B. Such a partition is impossible for the set $\{1, 2, 3, 4, 5\}$ and also for the set $\{1, 2, 3, 4, 5, 6\}$. Determine all values of $n$ for which the set of the first $n$ natural numbers can be partitioned into two subsets with no common elements such that the sum of the elements of each subset is the same.

Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \] [i]Laurentiu Panaitopol[/i]

Let $ABC$ be an equilateral triangle with circumcircle of center $O$ and radius $R$. Point $M$ is exterior to the triangle such that $S_bS_c = S_aS_b+S_aS_c$, where $S_a, S_b, S_c$ are the areas of triangles $MBC,MCA,MAB$ respectively. Prove that $OM \ge R$. Nguyễn Tiến Lâm

Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.

Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies: $$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$

Juty and Azgor plays the following game on a \((2n+1) \times (2n+1)\) board with Juty moving first. Initially all cells are colored white. On Juty's turn, she colors a white cell green and on Azgor's turn, he colors a white cell red. The game ends after they color all the cells of the board. Juty wins if all the green cells are connected, i.e. given any two green cells, there is at least one chain of neighbouring green cells connecting them (we call two cells [i]neighboring[/i] if they share at least one corner), otherwise Azgor wins. Determine which player has a winning strategy. [i]Proposed by Atonu Roy Chowdhury[/i]

Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.

There are $n^2$ empty boxes, each with a square base. The height and width of each box are integers between $1$ and $n$ inclusive, and no two boxes are identical. One box [i]fits inside[/i] another if its height and width are both smaller, and additionally, one of its dimensions is at least $2$ units smaller. In this way, we can form sequences of boxes (the first inside the second, the second inside the third, and so on). We place each of these sequences on a different shelf. How many shelves are needed to store all the boxes, with certainty?

Let $ABCD$ be a convex quadrilateral with $AB>AD$ and $\angle B=\angle D=90^{\circ}$. Let $P$ be a point in the side $AB$ such that $AP=AD$. The lines $PD$ and $BC$ cut in the point $Q$. The perpendicular line to $AC$ passing by $Q$ cuts $AB$ in the point $R$. Let $S$ be the foot of perpendicular of $D$ to the line $AC$. Prove that $\angle PSQ=\angle RCP$.

A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 18$

How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

Matilda drew $12$ quadrilaterals. The first quadrilateral is an rectangle of integer sides and $7$ times more width than long. Every time she drew a quadrilateral she joined the midpoints of each pair of consecutive sides with a segment. It´s is known that the last quadrilateral Matilda drew was the first with area less than $1$. What is the maximum area possible for the first quadrilateral? [asy]size(200); pair A, B, C, D, M, N, P, Q; real base = 7; real altura = 1; A = (0, 0); B = (base, 0); C = (base, altura); D = (0, altura); M = (0.5*base, 0*altura); N = (0.5*base, 1*altura); P = (base, 0.5*altura); Q = (0, 0.5*altura); draw(A--B--C--D--cycle); // Rectángulo draw(M--P--N--Q--cycle); // Paralelogramo dot(M); dot(N); dot(P); dot(Q); [/asy] $\textbf{Note:}$ The above figure illustrates the first two quadrilaterals that Matilda drew.

Solve in $ \mathbb R$ the equation $ \sqrt{1\minus{}x}\equal{}2x^2\minus{}1\plus{}2x\sqrt{1\minus{}x^2}$.

Triangle $ABC$ has side lengths $AB=18$, $BC=36$, and $CA=24$. The circle $\Gamma$ passes through point $C$ and is tangent to segment $AB$ at point $A$. Let $X$, distinct from $C$, be the second intersection of $\Gamma$ with $BC$. Moreover, let $Y$ be the point on $\Gamma$ such that segment $AY$ is an angle bisector of $\angle XAC$. Suppose the length of segment $AY$ can be written in the form $AY=\frac{p\sqrt{r}}{q}$ where $p$, $q$, and $r$ are positive integers such that $gcd(p, q)=1$ and $r$ is square free. Find the value of $p+q+r$.

Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

Let $ABC$ be an acute triangle. Variable points $E$ and $F$ are on sides $AC$ and $AB$ respectively such that $BC^2 = BA\cdot BF + CE \cdot CA$ . As $E$ and $F$ vary prove that the circumcircle of $AEF$ passes through a fixed point other than $A$ .

Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both $ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

Let ABC be a triangle with $AB = 8$, $BC = 9$, and $CA = 10$. The line tangent to the circumcircle of $ABC$ at $A$ intersects the line $BC$ at $T$, and the circle centered at $T$ passing through $A$ intersects the line $AC$ for a second time at $S$. If the angle bisector of $\angle SBA$ intersects $SA$ at $P$, compute the length of segment $SP$. [i]2016 CCA Math Bonanza Team #9[/i]

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$, $$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$

How many 3-digit numbers have the property that the sum of their digits is even?

Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.