[asy]draw((0,0)--(7,0)--(7,1)--(6,1)--(6,2)--(5,2)--(5,3)--(4,3)--(4,4)--(3,4)--(3,3)--(2,3)--(2,2)--(1,2)--(1,1)--(0,1)--(0,0)--cycle); draw((1,0)--(6,0)--(6,1)--(5,1)--(5,2)--(4,2)--(4,3)--(3,3)--(3,2)--(2,2)--(2,1)--(1,1)--(1,0)--cycle); draw((2,0)--(5,0)--(5,1)--(4,1)--(4,2)--(3,2)--(3,1)--(2,1)--cycle); draw((3,0)--(3,1)--(4,1)--(4,0)--cycle); fill((1,0)--(1,1)--(2,1)--(2,0)--cycle,black); fill((3,0)--(3,1)--(4,1)--(4,0)--cycle,black); fill((5,0)--(5,1)--(6,1)--(6,0)--cycle,black); fill((2,1)--(2,2)--(3,2)--(3,1)--cycle,black); fill((4,1)--(4,2)--(5,2)--(5,1)--cycle,black); fill((3,2)--(3,3)--(4,3)--(4,2)--cycle,black);[/asy] A "stair-step" figure is made up of alternating black and white squares in each row. Rows $ 1$ through $ 4$ are shown. All rows begin and end with a white square. The number of black squares in the $ 37$th row is \[ \textbf{(A)}\ 34 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 38 \]

In a football tournament $2n$ teams play in a round. Every round consists of $n$ pairs of teams that haven’t played with each other yet. Every round’s schedule is determined before the round is held. Find the minimal positive integer $k$ such that the following situation is possible: after $k$ rounds it’s impossible to schedule the next round.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions: $f(x) + f(y) \ge xy$ for all real $x, y$ and for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.

Three balloon vendors each offer two types of balloons - one offers red & blue, one offers blue & yellow, and one offers yellow & red. I like each vendor the same, so I must buy $7$ balloons from each. How many different possible triples $(x,y,z)$ are there such that I could buy $x$ blue, $y$ yellow, and $z$ red balloons?

In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.

What is the sum of the positive solutions to $2x^2 -\lfloor x \rfloor = 5$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$?

For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

$6$ musicians gathered at a chamber music festival. At each scheduled concert, some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that every $2$ musicians each must play for the other in some concert?

A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.

Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

Suppose $ a,c,d \in N$ and $ d|a^2b\plus{}c$ and $ d\geq a\plus{}c$ Prove that $ d\geq a\plus{}\sqrt[2b] {a}$

Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.) \[ x_i^2 + x_i - 1 = x_{i+1} \]

There are numerous sets of $17$ distinct positive integers that sum to $2018$, such that each integer has the same sum of digits in base $10$. Let $M$ be the maximum possible integer that could exist in any such set. Find the sum of $M$ and the number of such sets that contain $M$.

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?

Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.

If the ratio of the legs of a right triangle is $ 1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is: $ \textbf{(A)}\ 1: 4 \qquad \textbf{(B)}\ 1: \sqrt{2} \qquad \textbf{(C)}\ 1: 2 \qquad \textbf{(D)}\ 1: \sqrt{5} \qquad \textbf{(E)}\ 1: 5$

a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots. b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.

Jackie and Phil have two fair coins and a third coin that comes up heads with probability $ \frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $ \frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Consider two positive real numbers $ a,b $ and the function $ f:(0,\infty )\longrightarrow\left( \sqrt{ab} ,\frac{a+b}{2} \right) $ defined as $ f(x)=-x+\sqrt{x^2+(a+b)x+ab}. $ Prove that it's bijective. [i]D.M. Bătineți-Giurgiu[/i] and [i]Neculai Stanciu[/i]

The bottom surface of triangular pyramid $S-ABC$ is a regular triangle. Projection of $A$ on plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. If $H-AB-C=30^{\circ},SA=2\sqrt3$, then the volume of $S-ABC$ is________.

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.

The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.

Let $P$ be the set of all primes, and let $M$ be a non-empty subset of $P$. Suppose that for any non-empty subset ${p_1,p_2,...,p_k}$ of $M$, all prime factors of $p_1p_2...p_k+1$ are also in $M$. Prove that $M=P$. [i]Proposed by Alex Zhai[/i]