Let $a$, $b$, and $c$ be nonzero real numbers. Prove that $a+b+c$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ cannot both be $0$.

Let $n$ be a positive integer. We call $(a_1,a_2,\cdots,a_n)$ a [i]good[/i] $n-$tuple if $\sum_{i=1}^{n}{a_i}=2n$ and there doesn't exist a set of $a_i$s such that the sum of them is equal to $n$. Find all [i]good[/i] $n-$tuple. (For instance, $(1,1,4)$ is a [i]good[/i] $3-$tuple, but $(1,2,1,2,4)$ is not a [i]good[/i] $5-$tuple.)

Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.

Points $A,B,C,D,E$ lie in a counterclockwise order on a circle $O$, and $AC = CE$ $P=BD \cap AC$, $Q=BD \cap CE$ Let $O_1$ be the circle which is tangent to $\overline {AP}, \overline {BP}$ and arc $AB$ (which doesn't contain $C$) Let $O_2$ be the circle which is tangent $\overline {DQ}, \overline {EQ}$ and arc $DE$ (which doesn't contain $C$) Let $O_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X$ Prove that $XC$ bisects $\angle ACE$

If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$

The function $f(x) = x^2 + (2a + 3)x + (a^2 + 1)$ only has real zeroes. Suppose the smallest possible value of $a$ can be written in the form $p/q$, where $p, q$ are relatively prime integers. Find $|p| + |q|$.

Let $n$ be an integer with $n \geq 2$ and let $A \in \mathcal{M}_n(\mathbb{C})$ such that $\operatorname{rank} A \neq \operatorname{rank} A^2.$ Prove that there exists a nonzero matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $$AB=BA=B^2=0$$ [i]Cornel Delasava[/i]

Find the largest positive integer $n$ satisfying the following: [center] "There are precisely $53$ integers in the list of integers $1, 2, \ldots, n$ that are either perfect squares, perfect cubes or both."[/center]

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written $m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

In a rectangular $57\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells? [i]Proposed by James Lin

A strongman Bambula can carry several weights at the same time, if their total weight does not exceed $200$ kg, and these weights are no more than three. On the way to work, he injured his finger and found that he could now carry no more than two weights (and still no more than $200$ kg). At what minimum $k$ is the statement true: [i]any set of $100$ weights that Bambula could previously carry in $50$ runs, with a sore finger, he will be able to carry in no more than $k$ runs?[/i]

The $ 600$ students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of the three lunch groups. The probability that the three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately: \[ \textbf{(A)}\ \frac{1}{27} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{3} \]

Prove that it is impossible for polynomials $f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]$ to satisfy \[f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.\]

Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.

Consider subsets of the set $1 , 2,..., N$. For each such subset we can compute the product of the reciprocals of each member. Find the sum of all such products.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$. [i]Proposed by Luke Robitaille[/i]

Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ such that $\left|AC\right|\neq\left|BC\right|$. The internal angle bisector of $\angle CAB$ intersects side $BC$ at $D$ and the external angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $\Omega$ at $E$ and $F$ respectively. Let $G$ be the intersection of lines $AE$ and $FI$ and let $\Gamma$ be the circumcircle of triangle $BDI$. Show that $E$ lies on $\Gamma$ if and only if $G$ lies on $\Gamma$.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Points $A, B, C$ and $D$ lie on line $l$ in this order. Two circular arcs $C_1$ and $C_2$, which both lie on one side of line $l$, pass through points $A$ and $B$ and two circular arcs $C_3$ and $C_4$ pass through points $C$ and $D$ such that $C_1$ is tangent to $C_3$ and $C_2$ is tangent to $C_4$. Prove that the common external tangent of $C_2$ and $C_3$ and the common external tangent of $C_1$ and $C_4$ meet each other on line $l$. [i]Proposed by Ali Khezeli[/i]

Let the sequence of random variables $ \{ X_m, \; m \geq 0\ \}, \; X_0=0$, be an infinite random walk on the set of nonnegative integers with transition probabilities \[ p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \,\] \[ q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0.\] Prove that for arbitrary $ k >0$ there is an $ \alpha_k > 1$ such that \[ P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)\] satisfies the limit relation \[ \lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.\] [i]J. Tomko[/i]

Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]

For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }6 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad $

$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.