Find the number of positive integers $n$ less than $2017$ such that \[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \] is an integer.

A and B plays a game on a pyramid whose base is a $2016$-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarantee that all sides are colored.

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

Find all triples of non-negative integers \((a, b, c)\) such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]

Find all positive integers $n$ for such the following condition holds: "If $a$, $b$ and $c$ are positive integers such are all numbers \[ a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2 \] are divisible by $n$, then $(a+b+c)^2$ is also divisible by $n$." [i]G.M.Sharafetdinova[/i]

Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied: $$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$

How many ways are there to fill a $3 \times 3$ grid with the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$, such that the set of three elements in every row and every column form an arithmetic progression in some order? (Each number must be used exactly once)

The audience chooses two of five cards, numbered from $1$ to $5$ respectively. The assistant of a magician chooses two of the remaining three cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.

Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$ Prove that \[a_1^2+a_2^2+\ldots+a_{59}^2\le 2006\]

There are $12$ chairs arranged in a circle, numbered from $ 1$ to $ 12$. How many ways are there to select some of the chairs in such a way that our selection includes $3$ consecutive chairs somewhere?

Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.

A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.

A square of perimeter $20$ is inscribed in a square of perimeter $28$. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? $ \textbf{(A)}\ \sqrt{58} \qquad\textbf{(B)}\ \frac{7\sqrt{5}}{2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ \sqrt{65} \qquad\textbf{(E)}\ 5\sqrt{3} $

Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.

Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an [i]amazing[/i] number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets. a) Prove that every perfect square is amazing. b) Show that there are infinitely many positive integers which are not amazing.

13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, .. 13$. Their scores are $s_1,s_2, ... s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, ... w_5$. Each score $s_i$ is equal to the sums of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\frac{1}{\sum_ {s_i} }$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$. [i]Proposed by Jeff Lin[/i]

There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$?

Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\] Find $|f(0)|$.

Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres each with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r?$ $\textbf{(A) }\dfrac32\qquad\textbf{(B) }\dfrac{90 - 40\sqrt3}{11}\qquad\textbf{(C) }2\qquad\textbf{(D) }\dfrac{144 - 25\sqrt3}{44}\qquad\textbf{(E) }\dfrac52$

Do there exist infinitely many positive integers $a, b$ such that $$(a^2+1)(b^2+1)((a+b)^2+1)$$ is a perfect square? [i]Proposed Ivan Chan Guan Yu[/i]

([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).

A traffic light runs repeatedly through the following cycle: green for $ 30$ seconds, then yellow for $ 3$ seconds, and then red for $ 30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? $ \textbf{(A)}\ \frac {1}{63}\qquad \textbf{(B)}\ \frac {1}{21}\qquad \textbf{(C)}\ \frac {1}{10}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {1}{3}$

Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5? $ \text{(A)}\ \frac{1}{36}\qquad\text{(B)}\ \frac{1}{18}\qquad\text{(C)}\ \frac{1}{6}\qquad\text{(D)}\ \frac{11}{36}\qquad\text{(E)}\ \frac{1}{3} $

The circumcenter, circumradius and orthocenter of a triangle $ ABC $ satisfying $ AB<AC $ are notated with $ O,R,H, $ respectively. Prove that the middle of the segment $ OH $ belongs to the line $ BC $ if $$ AC^2-AB^2=2R\cdot BC. $$ [i]Marius Marchitan[/i]