For covering the floor of a rectangular room rectangular tiles of sizes $2 \times 2$ and $4 \times 1$ were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

How many three-digit numbers are perfect squares?

In Durer’s duck school, there are six rows of doors, as seen on the diagram; both rows are made up of three doors. Dodo duck wishes to enter the school from the street in a way that she uses all six doors exactly once. (On her path, she may go to the street again, or leave the school, so long as she finishes her path in the school.) How many ways can she perform this? [i]Two paths are considered different if Dodo takes the doors in a different order.[/i] [img]https://cdn.artofproblemsolving.com/attachments/5/1/8b722eb2c642e8275928753921fdfbd7495df9.png[/img]

Find all real values of x, y, and z such that $$x - \sqrt{yz} = 42$$ $$y - \sqrt{xz}=6$$ $$z-\sqrt{xy}=30$$

In the Cartesian plane, each point with integer coordinates is colored either red, green, or blue. It is possible to form right isosceles triangles ($45^\circ - 90^\circ - 45^\circ$) using colored points as vertices. Prove that regardless of how the coloring is done, there always exists a right isosceles triangle such that all its vertices are either the same color or all different colors.

Nine distinct digits appear in the decimal expansion of $2^{29}$. Which digit is missing?

It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\log 101$ and $\log 2018$. The last side of his triangle has side length $\log n$, where $n$ is an integer. How many possible values are there for $n$? [i]2018 CCA Math Bonanza Individual Round #6[/i]

Marjorie is the drum major of the world's largest marching band, with more than one million members. She would like the band members to stand in a square formation. To this end, she determines the smallest integer $n$ such that the band would fit in an $n \times n$ square, and lets the members form rows of $n$ people. However, she is dissatisfied with the result, since some empty positions remain. Therefore, she tells the entire first row of $n$ members to go home and repeats the process with the remaining members. Her aim is to continue it until the band forms a perfect square, but as it happens, she does not succeed until the last members are sent home. Determine the smallest possible number of members in this marching band.

A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box? $\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 144$

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. [i]Dan Schwarz[/i]

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $O$. The circumscribed circles of triangles $AOB$ and $COD$ intersect at point $M$ on the side $AD$. Prove that the point $O$ is the center of the inscribed circle of the triangle $BMC$.

Let $m>1$ be an odd integer. Let $n=2m$ and $\theta=e^{2\pi i\slash n}$. Find integers $a_{1},\ldots,a_{k}$ such that $\sum_{i=1}^{k}a_{i}\theta^{i}=\frac{1}{1-\theta}$.

A set $S$ of $100$ points, no four in a plane, is given in space. Prove that there are no more than $4 .101^2$ tetrahedra with the vertices in $S$, such that any two of them have at most two vertices in common.

Do there exist functions $p(x)$ and $q(x)$, such that $p(x)$ is an even function while $p(q(x))$
is an odd function (different from 0)? [i](3 points)[/i]

There are some checkers in $n\cdot n$ size chess board.Known that for all numbers $1\le i,j\le n$ if checkwork in the intersection of $i$ th row and $j$ th column is empty,so the number of checkers that are in this row and column is at least $n$.Prove that there are at least $\frac{n^2}{2}$ checkers in chess board.

Find the locus of the points of intersection of the altitudes of the triangles inscribed in a given circle.

All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center? $\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

What is the largest value of $t$ for which the solution of the problem \[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\sin x,\; u|_{t=0}=0\] can be extended to the interval $[0,t)$.

A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.

In a round-robin table tennis tournament between $2010$ athletes, where each match ends with a winner and a loser, let $a_1,... , a_{2010}$ denote the number of wins of each athlete, and let $b_1, .., b_{2010}$ denote the number of losses of each athlete. Show that $a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}$.

Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.

Solve in positive integers: $(x^{2}+2)(y^{2}+3)(z^{2}+4)=60xyz$.