High school graduate Igor Petrov, who dreamed of becoming a diplomat, took the entrance exam in mathematics to Moscow University. Igor remembered all the problems offered during the exam, but forgot some numerical data in one. This is the task: “When multiplying two natural numbers, the difference of which is $10$, an error was made: the hundreds digit in the product was increased by $2$. When dividing the resulting (incorrect) product by the smaller of the factors, the result was quotient $k$ and remainder $r$.. Find the numbers that needed to be multiplied.” . The values of $k$ and $r$ were given in the condition, but Igor forgot them. However, he remembered that the problem had two answers. What could the numbers $ k$ and $r$ be equal to (they are both integers and positive)? [i]Note. The problem in question was proposed at one of the humanities faculties of Moscow State University in 1991. [/i]

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$. [i]Proposed by Md. Ashraful Islam Fahim[/i]

Some of the people attending a meeting greet each other. Let $n$ be the number of people who greet an odd number of people. Prove that $n$ is even.

Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10} $

If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$, respectively, prove the inequality \[\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.\]

Find all positive numbers $x$ such that:$$\frac{1}{[x]}-\frac{1}{[2x]}=\frac{1}{6\{x\}}$$ where $[x]$ represents the integer part of $x$ and $\{x\}=x-[x]$.

Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$ ($D \in BC$), $M$ is the midpoint of $AD$ and $O$ is the circumcenter. Line $AO$ meets $BC$ at $K$ and circle of center $K$, radius $KA$ cuts $AB,AC$ at $E, F$ respectively. Prove that $AO$ bisects $EF$.

Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.

Prove that with $ n\ge 1 $ distinct numbers we can form an arithmetic progression if and only if there are exactly $ n-1 $ distinct elements in the set of positive differences between any two of these numbers.

Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?

Prove that the following inequality holds: \[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]

The sets $A_1,A_2,...,A_n$ are finite. With $d$ we denote the number of elements in $\bigcup_{i=1}^n A_i$ which are in odd number of the sets $A_i$. Prove that the number: $D(k)=d-\sum_{i=1}^n|A_i|+2\sum_{i<j}|A_i\cap A_j |+...+(-1)^k2^{k-1}\sum_{i_1<i_2<...<i_k}|A_{i_1}\cap A_{i_2}\cap ...\cap A_{i_k}|$ is divisible by $2^k$.

In a group of $2020$ people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends? [i]2020 CCA Math Bonanza Tiebreaker Round #1[/i]

Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of $n$, required to eliminate all the soldiers? [i]David Yang.[/i]

For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$. Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds. For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$ .

Point $P$ and line $\ell$ are such that the distance from $P$ to $\ell$ is $12$. Given that $T$ is a point on $\ell$ such that $PT = 13$, find the radius of the circle passing through $P$ and tangent to $\ell$ at $T$.

Given a natural number $ n \geq 2 $. Let $ a_1, a_2, \ldots , a_n $, $ b_1, b_2, \ldots , b_n $ be real numbers. Prove that the following conditions are equivalent: - For any real numbers $ x_1 \leq x_2 \leq \ldots \leq x_n $ holds the inequality $$\sum_{i=1}^n a_i x_i \leq \sum_{i=1}^n b_i x_i.$$ - For every natural number $ k\in \{1,2,\ldots, n-1\} $ holds the inequality $$ \sum_{i=1}^k a_i \geq \sum_{i=1}^k b_i, \ \ \text{ and } \\ \ \sum_{i=1}^n a_i = \sum_{i=1 }^n b_i.$$

Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$. The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$, determine the length of $YZ$

A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find $k$.

A sphere of radius $\sqrt{85}$ is centered at the origin in three dimensions. A tetrahedron with vertices at integer lattice points is inscribed inside the sphere. What is the maximum possible volume of this tetrahedron?

On each field of the board $ n\times n$ there is one figure, where $n\ge 2$. In one move we move every figure on one of its diagonally adjacent fields. After one move on one field there can be more than one figure. Find the least number of fields on which there can be all figures after some number of moves.

Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

Suppose that $X$ is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting $X$ with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.