Let $x,y,$ and $z$ be nonnegative real numbers with $x+y+z=120$. Compute the largest possible value of the median of the three numbers $2x+y,2y+z,$ and $2z+x$. [i]Proposed by Ankit Bisain[/i]

If $a=\tfrac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\arctan a + \arctan b$ equals $\textbf{(A) }\frac{\pi}{2}\qquad\textbf{(B) }\frac{\pi}{3}\qquad\textbf{(C) }\frac{\pi}{4}\qquad\textbf{(D) }\frac{\pi}{5}\qquad\textbf{(E) }\frac{\pi}{6}$

Using white cubes of side $1$, a prism (without holes) was assembled. The faces of the prism were painted black. It is known that the cubes left with exactly $4$ white faces are $20$ in total. Determine what the dimensions of the prism can be. Give all the possibilities.

In how many ways can we place the numbers from $1$ to $100$ in a $2\times 50$ rectangle (divided into $100$ unit squares) so that any two consecutive numbers are always placed in squares with a common side?

[b]Q12.[/b] Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively.

Joao had blue, white and red pearls and with them he made a necklace with $20$ pearls that has as many blue as white pearls. João noticed that, regardless of how he cut the necklace into two parts, both with an even number of pearls, one of the parts would always have more blue pearls than white ones. How many red pearls are in Joao's necklace?

Let $R$ be a ring of characteristic zero. Let $e,f,g\in R$ be idempotent elements (an element $x$ is called idempotent if $x^2=x$) satisfying $e+f+g=0$. Show that $e=f=g=0$.

A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.

Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1.$

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.

The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on the line BD is such that $|AB| = |AC| = |CD|.$ Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$ .

An urn initially contains one red ball and one blue ball. In each step, we choose a uniform random ball from the urn. If it is red, then another red ball and another blue ball are placed in the urn. And when we choose a blue ball for the $k$-th time, we put a blue ball and $2k + 1$ red balls in the urn. (The chosen balls are not removed; they remain in the urn.) Let $G_n$ denote the number of red balls in the urn after $n$ steps. Prove that there exist constants $0 < c, \alpha < \infty$ such that $\frac{G_n}{n^\alpha} \to c$ almost surely.

Positive integers $a, b, c$ satisfy $\frac1a +\frac1b +\frac1c<1$. Prove that $\frac1a +\frac1b +\frac1c\le \frac{41}{42}$. Also prove that equality in fact holds in the second inequality.

A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).

Let $P$ be a point in the interior of a triangle $ABC$ and let the rays $\overrightarrow{AP}, \overrightarrow{BP}$ and $\overrightarrow{CP}$ intersect the sides $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$, respectively. Let $D$ be the foot of the perpendicular from $A_1$ to $B_1C_1$. Show that \[\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.\]

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint? (A Shapovalov)

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

Let a, b be positive integers. a, b and a.b are not perfect squares. Prove that at most one of following equations $ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$ has solutions in positive integers.

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$. Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.

Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?

The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).

Let $n$ be an odd integer. Placing no more than one $X$ in each cell of a $n \times n$ grid, what is the greatest number of $X$ 's that can be put on the grid without getting $n$ $X$'s together vertically, horizontally or diagonally? [list=1] [*] $2{{n}\choose {2}}$ [*] ${{n}\choose {2}}$ [*] $2n $ [*] $2{{n}\choose {2}}-1$ [/list]