A government’s land with dimensions $n \times n$ are going to be sold in phases. The land is divided into $n^2$ squares with dimension $1 \times 1$. In the first phase, $n$ farmers bought a square, and for each rows and columns there is only one square that is bought by a farmer. After one season, each farmer could buy one more square, with the conditions that the newly-bought square has a common side with the farmer’s land and it hasn’t been bought by other farmers. Determine all values of n such that the government’s land could not be entirely sold within $n$ seasons.

Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$.

There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed: -the first card (from the top) is put in the bottom of the deck. -the second card (from the top) is taken away of the deck. -the third card (from the top) is put in the bottom of the deck. -the fourth card (from the top) is taken away of the deck. - ... The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.

How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions \[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\] for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?

The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?

Among a set of $32$ coins , all identical in appearance, $30$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight , which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times? (A Shapovalov)

Let $n$ be a positive integer. For a set of points in the plane $M$, we call $2$ distinct points $A,B \in M$ [i]connected[/i] if the line $AB$ contains exactly $n+1$ points from $M$. Find the minimum value of a positive integer $m$ such that there exists a set $M$ of $m$ points in the plane with the property that any point $A \in M$ is connected with exactly $2n$ other points from $M$.

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $ f(x) \equal{} ax^{2} \plus{} bx \plus{} c$ are equal. Their common value must also be which of the following? $ \textbf{(A)}\ \text{the coefficient of }x^{2}\qquad \textbf{(B)}\ \text{the coefficient of }x$ $ \textbf{(C)}\ \text{the y \minus{} intercept of the graph of }y \equal{} f(x)$ $ \textbf{(D)}\ \text{one of the x \minus{} intercepts of the graph of }y \equal{} f(x)$ $ \textbf{(E)}\ \text{the mean of the x \minus{} intercepts of the graph of }y \equal{} f(x)$

A toboggan sled is traveling at $\text{2.0 m/s}$ across the snow. The sled and its riders have a combined mass of $\text{120 kg}$. Another child ($m_{\text{child}} = \text{40 kg}$) headed in the opposite direction jumps on the sled from the front. She has a speed of $\text{5.0 m/s}$ immediately before she lands on the sled. What is the new speed of the sled? Neglect any effects of friction. (a) $\text{0.25 m/s}$ (b) $\text{0.33 m/s}$ (c) $\text{2.75 m/s}$ (d) $\text{3.04 m/s}$ (e) $\text{3.67 m/s}$

Let $\clubsuit\left(x\right)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit\left(8\right)=8$ and $\clubsuit\left(123\right)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit\left(\clubsuit\left(x\right)\right)=3$?

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve? [asy] size(170); defaultpen(fontsize(6pt)); dotfactor=4; label("$\circ$",(0,1)); label("$\circ$",(0.865,0.5)); label("$\circ$",(-0.865,0.5)); label("$\circ$",(0.865,-0.5)); label("$\circ$",(-0.865,-0.5)); label("$\circ$",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy] $ \textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 $

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Compute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.

Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.

In triangle $ABC$, $\angle ABC = 54^\circ$ and $\angle ACB = 42^\circ$. Point $D$ is the foot of the altitude from vertex $A$ to $BC$, and $I$ is the incenter of $\triangle ABC$. Point $K$ lies on line $AD$, such that $D$ is between $A$ and $K$ and $AK$ is equal to the diameter of the circumcircle of $\triangle ABC$. Find the measure of $\angle KID$.

Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten. [i]Proposed by Evan Chen[/i]

In $\triangle ABC$, prove that $1< \sum_{cyc}{\cos A}\le \frac{3}{2}$.

Let $n$ be a positive integer. (a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$. (b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.

We say that a [i]special word[/i] is any sequence of letters [b]containing a vowel[/b]. How many ordered triples of special words $(W_1,W_2,W_3)$ have the property that if you concatenate the three words, you obtain a rearrangement of "aadvarks"? For example, the number of triples of special words such that the concatenation is a rearrangement of ``adaa" is $6$, and all of the possible triples are: [center] (da,a,a),(ad,a,a),(a,da,a),(a,ad,a),(a,a,da),(a,a,ad). [/center] [i]2021 CCA Math Bonanza Team Round #5[/i]

There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.

For $2n$ positive integers a matching (i.e. dividing them into $n$ pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least $n!$ non-square matchings. (By $n!$ denote the product $1\cdot 2\cdot 3\cdot \ldots \cdot n$.)

Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.

Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$: \[1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.\]

Find all pairs of integers $a$ and $b$ for which \[7a+14b=5a^2+5ab+5b^2\]

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent. [i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]