In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions. [list][*]If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s. [*]If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.[/list] Find all pairs $(a,b)$ for which this is possible.

Let $\Omega$ be a circle with center $O$. Let $\omega_1$ and $\omega_2$ be circles with centers $O_1$ and $O_2$, respectively, internally tangent to $\Omega$ at points $A$ and $B$, respectively, such that $O_1$ is on $\overline{OA}$, and $O_2$ is on $\overline{OB}$ and $\omega_1$. There exists a point $P$ on line $AB$ such that $P$ is on both $\omega_1$ and $\omega_2$. Let the external tangent of $\omega_1$ and $\omega_2$ on the same side of line $AB$ as $O$ hit $\omega_1$ at $X$ and $\omega_2$ at $Y$, and let lines $AX$ and $BY$ intersect at $N$. Given that $O_1X = 81$ and $O_2Y = 18$, the value of $NX \cdot NA$ can be written as $a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find $a+b+c$. [i]Proposed by Kevin Zhao[/i]

Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that \[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]

Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

There are $256$ lumps of metal that have different weights in pairs. With the help of a beam balance , one may now compare every two lumps. Find the smallest number $m$ such that you can be sure to find the heaviest as well as the lightest lump with the weighing process.

Let $ a_1 , a_2 , \cdots , a_k $ be numbers such that $ a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k) $. Let $ z = ( x_k , x_{k-1} , \cdots , x_1 )_4 $ be a base 4 expansion of $ z \in \{ 0, 1, 2, \cdots , 4^k -1 \} $. Define $ A $ as follows: \[ A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}\] where \[ p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} . \] Prove that the number of elements in $ X $ is a power of 2.

Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$

Let $f(x)=2x^3-2x$. For what positive values of $a$ do there exist distinct $b,c,d$ such that $(a,f(a)),(b,f(b)),(c,f(c)),(d,f(d))$ is a rectangle?

A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$. Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola. [b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.

On the market one seller is selling watermelons, melons and young corn cobs. Total number of watermelons, melons and corn cobs is $239$. One buyer bought $\frac{2}{3}$ of all watermelons, $\frac{3}{5}$ of all melons and $\frac{5}{7}$ of all corn cobs. Other buyer bought $\frac{1}{13}$ of all watermelons, $\frac{1}{4}$ of all melons and $\frac{1}{5}$ of all corn cobs. How many pieces in total bought second buyer and how many seller had at the beggining of each watermelons, melons and corn cobs?

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.