At $ 2: 15$ o'clock, the hour and minute hands of a clock form an angle of: $ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 5^{\circ} \qquad\textbf{(C)}\ 22\frac {1}{2}^{\circ} \qquad\textbf{(D)}\ 7\frac {1}{2} ^{\circ} \qquad\textbf{(E)}\ 28^{\circ}$

Specified natural number $a$. Prove that there are an infinite number of prime numbers $p$ such that for some natural $n$ the number $2^{2^n} + a$ is divisible by $p$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Given $m, n$ such that $m > n^{n-1}$ and the number $m+1$, $m+2$,$ ...$, $m+n$ are composite. Prove that there exist distinct primes $p_1, p_2, ..., p_n$ such that $m + k$ is divisible by $p_k$ for each $k = 1, 2, ...$

Which properties of electromagnetic radiation are inversely related? $ \textbf{(A)}\ \text{amplitude and frequency} \qquad$ $\textbf{(B)}\ \text{energy and wavelength} \qquad$ $\textbf{(C)}\ \text{energy and frequency} \qquad$ $\textbf{(D)}\ \text{wavelength and amplitude}\qquad$

On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.

If two poles $ 20''$ and $ 80''$ high are $ 100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is: $ \textbf{(A)}\ 50'' \qquad\textbf{(B)}\ 40'' \qquad\textbf{(C)}\ 16'' \qquad\textbf{(D)}\ 60'' \qquad\textbf{(E)}\ \text{none of these}$

Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.) [i]Proposed by Ankan Bhattacharya, USA[/i]

Let $n$ be the number of isosceles triangles whose vertices are also the vertices of a regular 2019-gon. Then the remainder when $n$ is divided by 100 [list=1] [*] 15 [*] 25 [*] 35 [*] 65 [/list]

Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the first time the sum of the squares of the numbers was not changed find how this sum was changed the second time.

Find all positive integers $n$ such that \[\cos\frac{\pi}{n}\cos\frac{2\pi}{n}\cos\frac{3\pi}{n}=\frac{1}{n+1}\]

For a complex number $ z\equal{}x\plus{}iy$ we denote by $ P(z)$ the corresponding point $ (x,y)$ in the plane. Suppose $ z_1,z_2,z_3,z_4,z_5,\alpha$ are nonzero complex numbers such that: $ (i)$ $ P(z_1),...,P(z_5)$ are vertices of a complex pentagon $ Q$ containing the origin $ O$ in its interior, and $ (ii)$ $ P(\alpha z_1),...,P(\alpha z_5)$ are all inside $ Q$. If $ \alpha\equal{}p\plus{}iq$ $ (p,q \in \mathbb{R})$, prove that $ p^2\plus{}q^2 \le 1$ and $ p\plus{}q \tan \frac{\pi}{5} \le 1$.

Dilhan has objects of $3$ types, $A$, $B$, and $C$, and $6$ functions $$f_{A,B},f_{A,C},f_{B,A},f_{B,C},f_{C,A},f_{C,B}$$where $f_{X,Y}$ takes in an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his $6$ functions, without repeats, such that the resulting expression is well-typed, meaning an object can be taken in by the first function, and the resulting output can then be taken in by the second function, and so on. In how many orders can he compose his $6$ functions, satisfying this constraint? [i]Proposed by Adam Bertelli[/i]

Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $B$ at a speed of $7$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie? [asy] defaultpen(linewidth(0.8)); draw((100,0)--origin--60*dir(60), EndArrow(5)); label("$A$", origin, SW); label("$B$", (100,0), SE); label("$100$", (50,0), S); label("$60^\circ$", (15,0), N);[/asy]

Let $ABC$ be a triangle with incenter $I$, and let the tangency points of its incircle with its sides $AB$, $BC$, $CA$ be $C'$, $A'$ and $B'$ respectively. Prove that the circumcenters of $AIA'$, $BIB'$, and $CIC'$ are collinear.

Let $n$ be a natural number and $\alpha ,\beta ,\gamma$ be the angles of an acute triangle. Determine the least possible value of the sum: $T=tan^n \alpha+tan^n \beta+tan^n \gamma$.

Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\] [i]P. Medgyessy[/i]

Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?

The [i]popularity [/i] of a positive integer $n$ is the number of positive integer divisors of $n$. For example, $1$ has popularity $1$, and $12$ has popularity $6$. For each number $n$ between $1$ and $30$ inclusive, Cathy writes the number $n$ on $k$ pieces of paper, where $k$ is the popularity of $n$. Cathy then picks a piece of paper at random. Compute the probability that she will pick an even integer.

Let $P(x)$ be a polynomial with integer coefficients. We will denote the set of all prime numbers by $\mathbb P$. Show that the set $\mathbb S := \{p\in\mathbb P : \exists\text{ }n \text{ s.t. }p\mid P(n)\}$ is finite if and only if $P(x)$ is a non-zero constant polynomial.

Let $ S$ be a set of rational numbers such that (a) $ 0\in S;$ (b) If $ x\in S$ then $ x\plus{}1\in S$ and $ x\minus{}1\in S;$ and (c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x\minus{}1)}\in S.$ Must $ S$ contain all rational numbers?

Find the sum of all positive integers $a$ such that there exists an integer $n$ that satisfies the equation: \[a! \cdot 2^{\lfloor \sqrt{a} \rfloor}=n!.\] [i]Proposed by Ivy Zheng[/i]

Let $ A$, $ M$, and $ C$ be digits with \[ (100A \plus{} 10M \plus{} C )(A \plus{} M \plus{} C ) \equal{} 2005. \]What is $ A$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$

Find the dimension of the solution space of the problem $\partial u/\partial\overline{z} = a\delta(z —-i) + b\delta(z + i)$ for $|z|\le 2$, $\text{Im } u = 0$ for $|z| = 2$.