A young baseball player thinks he has hit a home run and gets excited, but, instead, he has just hit it to an outfielder who is just able to catch the ball, and does so at ground level. The ball was hit at a height of 1.5 meters from the ground at an angle $\phi$ above the horizontal axis. The catch was taken at a horizontal distance 30 meters from home plate, which was where the batter hit the ball. The ball left the bat at a speed of 21 m/s. Find all possible values $0<\phi<90^\circ$, in degrees, rounded to the nearest integer. You may use WolframAlpha, Mathematica, or a graphing aid to compute $\phi$ after you derive an expression to solve for it. [i](Proposed by Ahaan Rungta)[/i]

The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum.

We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

Real numbers $x,y,z$ satisfy the inequalities $$x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.$$Find the minimum and maximum possible values of $z$.

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

Let $n$ be a positive integer. Prove that $$\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ ...+\frac{a_n}{b_n} \ge n $$ where $(b_1, b_2, ..., b_n)$ is any permutation of the positive real numbers $a_1, a_2, ..., a_n$.

Evariste has drawn twelve triangles as follows, so that two consecutive triangles share exactly one edge. [img]https://cdn.artofproblemsolving.com/attachments/6/2/50377e7ad5fb1c40e36725e43c7eeb1e3c2849.png[/img] Sophie colors every triangle side in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?

Two line $BC$ and $EF$ are parallel. Let $D$ be a point on segment $BC$ different from $B$,$C$. Let $I$ be the intersection of $BF$ ans $CE$. Denote the circumcircle of $\triangle CDE$ and $\triangle BDF$ as $K$,$L$. Circle $K$,$L$ are tangent with $EF$ at $E$,$F$,respectively. Let $A$ be the other intersection of circle $K$ and $L$. Let $DF$ and circle $K$ intersect again at $Q$, and $DE$ and circle $L$ intersect again at $R$. Let $EQ$ and $FR$ intersect at $M$.\\ Prove that $I$, $A$, $M$ are collinear.

The sum of the two $ 5$-digit numbers $ AMC10$ and $ AMC12$ is $ 123422$. What is $ A\plus{}M\plus{}C$? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

Show that there do not exist non-zero real numbers $a,b,c$ such that the following statements hold simultaneously: $\bullet$ the equation $ax^2 + bx + c = 0$ has two distinct roots $x_1,x_2$; $\bullet$ the equation $bx^2 + cx + a = 0$ has two distinct roots $x_2,x_3$; $\bullet$ the equation $cx^2 + ax + b = 0$ has two distinct roots $x_3,x_1$. (Note that $x_1,x_2,x_3$ may be real or complex numbers.)

Kelvin the Frog writes 2020 words on a blackboard, with each word chosen uniformly randomly from the set $\{\verb|happy|, \verb|boom|, \verb|swamp|\}$. A multiset of seven words is [i]merry[/i] if its elements can spell $``\verb|happy happy boom boom swamp swamp swamp|."$ For example, the eight words \[\verb|swamp|, \verb|happy|, \verb|boom|, \verb|swamp|, \verb|swamp|, \verb|boom|, \verb|swamp|, \verb|happy|\] contain four merry multisets. Determine the expected number of merry multisets contained in the words on the blackboard. [size=6][url]http://www.hpmor.com/chapter/12[/url][/size]

The cities of Terra Brasilis are connected by some roads. There are no two cities directly connected by more than one road. It is known that it is possible to go from one city to any other using one or more roads. We call [i]role[/i] any closed road route (ie, it starts in a city and ends in the same city) that does not pass through a city more than once. At Terra Brasilis, all roles go through an odd number of cities. The government of Terra Brasilis decided to close some roles for reform. When you close a role, all it;s roads are closed, so traffic is not allowed on these roads. By doing this, Terra Brasilis was divided into several regions such that from any city in each region it is possible to reach any other in the same region by road, but it is not possible to reach cities in other regions. Prove that the number of regions is odd [hide=original wording]As cidades da Terra Brasilis sao conectadas por algumas estradas. Nao ha duas cidades conectadas diretamente por mais de uma estrada. Sabe-se que, e possivel ir de uma cidade para qualquer outra utilizando uma ou mais estradas. Chamamos de rol^e qualquer rota fechada de estradas (isto e, comeca em uma cidade e termina na mesma cidade) que nao passa por uma cidade mais de uma vez. Na Terra Brasilis, todos os roles passam por quantidades impares de cidades. O governo da Terra Brasilis decidiu fechar alguns roles para reforma. Ao fechar um role, todas as suas estradas sao interditadas, de modo que nao e permitido o trafego nessas estradas. Ao fazer isso, a Terra Brasilis ficou dividida em varias regioes de modo que, de qualquer cidade de cada regiao e possivel hegar a qualquer outra da mesma regiao atraves de estradas, mas nao e possivel hegar a cidades de outras regioes. Prove que o numero de regioes e impar.[/hide]

In the triangle $ABC$, ${{A}_{1}}$ and ${{C}_{1}} $ are the midpoints of sides $BC $ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$. Prove that $\angle BP {{A}_{1}} = \angle PAC $.

Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$ Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$ [i]Proposed by Ma Zhao Yu

Prove that there doesn't exist any positive integer $n$ such that $2n^2+1,3n^2+1$ and $6n^2+1$ are perfect squares.

Let $f$ be defined as below for integers $n \ge 0$ and $a_0, a_1, ...$ such that $\sum_{i\ge 0}a_i$ is finite: $$f(n; a_0, a_1, ...) = \begin{cases} a_{2020}, & \text{ $n = 0$} \\ \sum_{i\ge 0} a_i f(n-1;a_0,...,a_{i-1},a_i-1,a_{i+1}+1,a_{i+2},...)/ \sum_{i\ge 0}a_i & \text{$n > 0$} \end{cases}$$. Find the nearest integer to $f(2020^2; 2020, 0, 0, ...)$.

There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$, for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

Circle $I$ is the incircle of $\triangle ABC$. Circle $I$ is tangent to sides $BC$ and $AC$ at $M,N$ respectively. $E,F$ are midpoints of sides $AB$ and $AC$ respectively. Lines $EF, BI$ intersect at $D$. Show that $M,N,D$ are collinear.

Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.

Let $K, L, M$ denote three points on the sides $BC$, $AB$ and $BC$ of $\triangle{ABC}$, so that $ALKM$ is a parallelogram. Points $S$ and $T$ are chosen on lines $KL$ and $KM$ respectively, so that the quadrilaterals $AKBS$ and $AKCT$ are both cyclic. Prove that $MLST$ is cyclic if and only if $K$ is the midpoint of $BC$.

Let $O$ be the center of the equilateral triangle $ABC$. Pick two points $P_1$ and $P_2$ other than $B$, $O$, $C$ on the circle $\odot(BOC)$ so that on this circle $B$, $P_1$, $P_2$, $O$, $C$ are placed in this order. Extensions of $BP_1$ and $CP_1$ intersects respectively with side $CA$ and $AB$ at points $R$ and $S$. Line $AP_1$ and $RS$ intersects at point $Q_1$. Analogously point $Q_2$ is defined. Let $\odot(OP_1Q_1)$ and $\odot(OP_2Q_2)$ meet again at point $U$ other than $O$. Prove that $2\,\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ$. Remark. $\odot(XYZ)$ denotes the circumcircle of triangle $XYZ$.

The number of positive integers $k$ for which the equation $kx - 12 = 3k$ has an integer solution for $x$ is $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7 $

Exactly one integer is written in each square of an $n$ by $n$ grid, $n \ge 3$. The sum of all of the numbers in any $2 \times 2$ square is even and the sum of all the numbers in any $3 \times 3$ square is even. Find all $n$ for which the sum of all the numbers in the grid is necessarily even.