Zeros and ones are placed in each square of a rectangular board. Such a board is said to be [i]varied[/i] if each row contains at least one $0$ and at least two $1$s. Given n$\geq 3,$ find all integers $k>1$ with the following property: The columns of each varied board of $k$ rows and n columns can be permuted so that in each row of the new board the $1$s do not form a block (that is, there are at least two $1$s that are separated by one or more $0$s).

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

The sidelengths of a polygon with $1994$ sides are $a_{i}=\sqrt{i^2 +4}$ $ \; (i=1,2,\cdots,1994)$. Prove that its vertices are not all on lattice points.

In a triangle $ABC$, let $P$ be a point on the bisector of $\angle BAC$ and let $A',B'$ and $C'$ be points on lines $BC,CA$ and $AB$ respectively such that $PA'$ is perpendicular to $BC,PB'\perp AC$, and $PC'\perp AB$. Prove that $PA'$ and $B'C'$ intersect on the median $AM$, where $M$ is the midpoint of $BC$.

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]

Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.

We have tiles (which are build from squares of side length 1) of following shapes: [asy] unitsize(0.5 cm); draw((1,0)--(2,0)); draw((1,1)--(2,1)); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((0,1)--(1,1)); draw((0,2)--(1,2)); draw((0,1)--(0,2)); draw((1,1)--(1,2)); draw((0, 0)--(1, 0)); draw((0, 0)--(0, 1)); draw((5,0)--(6,0)); draw((5,1)--(6,1)); draw((5,0)--(5,1)); draw((6,0)--(6,1)); draw((4,1)--(5,1)); draw((5,2)--(6,2)); draw((5,1)--(5,2)); draw((6,1)--(6,2)); draw((4, 0)--(5, 0)); draw((4, 0)--(4, 1)); draw((6,2)--(7,2)); draw((7,1)--(7,2)); draw((6,1)--(7,1)); draw((11,0)--(12,0)); draw((11,1)--(12,1)); draw((11,0)--(11,1)); draw((12,0)--(12,1)); draw((10,1)--(11,1)); draw((10,2)--(11,2)); draw((10,1)--(10,2)); draw((11,1)--(11,2)); draw((10, 0)--(11, 0)); draw((10, 0)--(10, 1)); draw((9, 2)--(9, 1)); draw((9,1)--(10, 1)); draw((9,2)--(10,2)); [/asy] For each odd integer $n \ge 7$, determine minimal number of these tiles needed to arrange square with side of length $n$. (Attention: Tiles can be rotated, but they can't overlap.)

Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$. Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

In how many ways can we construct a square with dimensions $4\times 4$ using $4$ white, $4$ green , $4$ red and 4 $blue$ squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .

In a rectangular array of nonnegative reals with $m$ rows and $n$ columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that $m=n$.

The points $A_1, B_1, C_1$ are chosen on the sides $BC, CA, AB$ of a triangle $ABC$ so that $BA_1=BC_1$ and $CA_1=CB_1$. The lines $C_1A_1$ and $A_1B_1$ meet the line through $A$, parallel to $BC$, at $P, Q$. Let the circumcircles of the triangles $APC_1$ and $AQB_1$ meet at $R$. Given that $R$ lies on $AA_1$, show that $R$ lies on the incircle of $ABC$.

Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

In a party of $2020$ people, some pairs of people are friends. We say that a given person's [i]popularity[/i] is the size of the largest group of people in the party containing them with the property that every pair of people in that group is friends. A person has popularity number $1$ if they have no friends. What is the largest possible number of distinct popularities in the party? [i]2021 CCA Math Bonanza Tiebreaker Round #3[/i]

10.8 Suppose $S$ is a set of points in the plane, $|S|$ is even; no three points of $S$ are collinear. Prove that $S$ can be partitioned into two sets $S_1$ and $S_2$ so that their convex hulls have equal number of vertices.

[b]4.[/b] Let $x_1 , x_2 , y_1 , y_2 , z_1 , z_2 $ be transcendental numbers. Suppose that any 3 of them are algebraically independent, and among the 15 four-tuples on $\{x_1 , x_2 , y_1, y_2 \}$, $\{ x_1 , x_2 , z_1 , z_2 \} $ and $ \{y_1 , y_2 , z_1 , z_2 \} $ are algebraically dependent. Prove that there exists a transcendental number $t$ that depends algebraically on each of the pairs $\{ x_1 , x_2\}$ , $\{ y_1 , y_2 \}$, and $\{ z_1 , z_2 \}$. ([b]A.37[/b]) [L. Lovász]

Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$. Calculate the area of the quadrilateral.

Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients. [i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]

Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}} \ge 6r$ (D.Shvetsov)

Let $r,s,u,v$ be real numbers. Prove that: $$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$

There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.