A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step? $ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$

Consider all 1000-element subsets of the set $\{1,2,3,\dots,2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? $\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64 $

On a board, Ana writes $a$ different integers, while Ben writes $b$ different integers. Then, Ana adds each of her numbers with with each of Ben’s numbers and she obtains $c$ different integers. On the other hand, Ben substracts each of his numbers from each of Ana’s numbers and he gets $d$ different integers. For each integer $n$ , let $f(n)$ be the number of ways that $n$ may be written as sum of one number of Ana and one number of Ben. [i]a)[/i] Show that there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{c}.$$ [i]b)[/i] Does there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{d}?$$ [i]Proposed by Besfort Shala, Kosovo[/i]

The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is: $\textbf{(A)}\ 33\qquad \textbf{(B)}\ 34\qquad \textbf{(C)}\ 35\qquad \textbf{(D)}\ 100\qquad \textbf{(E)}\ \text{none of these}$

$$Problem 2 :$$If ABCD is as convex quadrilateral with $\angle ADC = 30$ and $BD = AB+BC+CA$, prove that $BD$ bisects $\angle ABC$.

Each of $4$ stones weights the integer number of grams. A balance with arrow indicates the difference of weights on the left and the right sides of it. Is it possible to determine the weights of all stones in $4$ weighings, if the balance can make a mistake in $1$ gram in at most one weighing?

Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.

For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then \[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \] equals $\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$

Let $ABCDE$ be a convex pentagon that satisfies all of the following:[list] [*]There is a circle $\Gamma$ tangent to each of the sides. [*]The lengths of the sides are all positive integers. [*]At least one of the sides of the pentagon has length $1$. [*]The side $AB$ has length $2$.[/list] Let $P$ be the point of tangency of $\Gamma$ with $AB$.[list] (a) Determine the lengths of the segments $AP$ and $BP$. (b) Give an example of a pentagon satisfying the given conditions.[/list]

Any circle in the $xy$-plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle. Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid. By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.

A sequence with first term $a_0$ is defined such that $a_{n+1}=2a_n^2-1$ for $n\geq0.$ Let $N$ denote the number of possible values of $a_0$ such that $a_0=a_{2020}.$ Find the number of factors of $N.$ [i]Proposed by Alex Li[/i]

Given a triangle with sides $a, b, c$ and medians $s_a, s_b, s_c$ respectively. Prove the following inequality: $$a + b + c> s_a + s_b + s_c> \frac34 (a + b + c) $$

Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and for all points $X \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides

Evaluate \[2\times (2\times (2\times (2\times (2\times (2\times 2-2)-2)-2)-2)-2)-2.\] [i]Proposed by Nathan Xiong[/i]

How many subsets of $\{1, 2, \cdots, 10\}$ are there that don't contain $2$ consecutive integers?

Compute the number of values of $x$ in the interval $[-11\pi,-2\pi]$ that satisfy $\tfrac{5\cos(x)+4}{5\sin(x)+3}=0$.

Let $p$ be a prime. Find all positive integers $k$ such that the set $\{1,2, \cdots, k\}$ can be partitioned into $p$ subsets with equal sum of elements.

Compute the number of positive integers $n\leq1330$ for which $\tbinom{2n}{n}$ is not divisible by $11$.

Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements

Thin and Fat eat a pizza of $2n$ pieces. Each piece contains a distinct amount of olives between $1$ and $2n$. Thin eats the first piece, and the two players alternately eat a piece neighbor of an eaten piece. However, neither Thin nor Fat like olives, so they will choose pieces that minimizes the total amount of olives they eat. For each arrangement $\sigma$ of the olives, let $s(\sigma)$ the minimal amount of olives that Thin can eat, considering that both play in the best way possible. Let $S(n)$ the maximum of $s(\sigma)$, considering all arrangements. $a)$ Prove that $n^2-1+\lfloor \frac{n}{2} \rfloor \le S(n) \le n^2+\lfloor \frac{n}{2} \rfloor$ $b)$ Prove that $S(n)=n^2-1+\frac{n}{2}$ for each even n.

On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.

Four heaps contain $38,45,61$ and $70$ matches respectively. Two players take turn choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy ?

Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.

One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has $\{1, 2, 3, 6, 9\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow? [asy] path p=origin--(18,0)--(18,5)--(0,5)--cycle; draw(p^^shift(0,8)*p^^shift(22,0)*p^^shift(22,16)*p^^shift(22,24)*p); filldraw(shift(0,16)*p^^shift(22,8)*p^^shift(22,32)*p^^shift(0,32)*p^^shift(0,24)*p, black, black); draw((-1,-1)--(41,-1)--(41,38)--(-1,38)--cycle, linewidth(2)); int i; for(i=1; i<6; i=i+1) { label(string(6-i), (-3,8*i-5.5), W); label(string(11-i), (43,8*i-5.5), E); }[/asy]