Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.

Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? (Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.) [i]Proposed by Michael Tang[/i]

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

In a triangle $\triangle ABC$ with orthocenter $H$, let $BH$ and $CH$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. If the tangent line to the circumcircle of $\triangle ABC$ passing through $A$ intersects $BC$ at $P$, $M$ is the midpoint of $AH$, and $EF$ intersects $BC$ at $G$, then prove that $PM$ is parallel to $GH$. [i]Proposed by Sreejato Bhattacharya[/i]

All sides of triangle $ABC$ are different. On rays $B A$ and $C A$ the segments $B K$ and $CM$ are laid out, equal to side $BC$. Let us denote by $x$ the length of the segment $KM$. In the same way, by plotting the side $AC$ on the rays $AB$ and $CB$ from $A$ and $C$, we obtain a segment of length $y$, and by plotting the side AB on the rays $AC$ and $BC$, we obtain a segment of length $z$. a) Prove that a triangle can be formed from the segments $x$, $y$ and $z$, and this triangle is similar to triangle $ABC$. b) Find the radius of the circumcircle of a triangle with sides $x$, $y$ and $z$, if the radii of the circumscribed and inscribed circles of triangle $ABC$ are equal to $R$ and $r$ respectively.

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$: $$0\le y+f(x)-f^{f(y)}(x)\le1$$ that here $$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$

There is a family of curves: $P_0,P_1,P_2,\cdots$. $P_0$ is a regular triangle, whose area is $1$. For all $k\in\mathbb{Z}_+$, $P_k$ is defined in this way: trisect all sides of $P_{k-1}$, and draw outward a regular triangle with side of the segment in the middle, then cut off the segment in the middle. $S_n$ is the area of $P_n$. [b](a)[/b] Find $S_n$. [b](b)[/b] Find $\lim_{n\to\infty}S_n$.

Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if \[\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} \]

On a \( 10 \times 10 \) chessboard, there are 91 white pawns placed in different squares. Nico picks a white pawn, paints it black, and places it in an empty square, repeating the process until all pawns have been painted. Prove that at some point, there will be two pawns of different colors placed on squares that share a common edge.

[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$. The incircle of triangle $ABC$ is tangent to the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D,E,F$ respectively. Let $M$ be the midpoint of $\overline{EF}$. Let $P$ be the projection of $A$ onto $BC$ and let $K$ be the intersection of $MP$ and $AD$. Prove that the circumcircles of triangles $AFE$ and $PDK$ have equal radius. [i]Kyprianos-Iason Prodromidis[/i]

Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.

Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $.0001$ of the correct answer will be accepted.

Find the minimum and maximum value of the function \begin{align*} f(x,y)=ax^2+cy^2 \end{align*} Under the condition $ax^2-bxy+cy^2=d$, where $a,b,c,d$ are positive real numbers such that $b^2 -4ac <0$

Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.

Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$ i) $f(1)=2016$ ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$

Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

A clockmaker wants to design a clock such that the area swept by each hand (second, minute, and hour) in one minute is the same (all hands move continuously). What is the length of the hour hand divided by the length of the second hand?

Does there exist a triangle such that the lengths of all its sides and altitudes are integers and its perimeter is equal to $1995$?

Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit number with all digits equal.

Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and find the ratio of their areas.

Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.