Among $18$ parts placed in a row, some three in a row weigh $99 $ g each, and all the rest weigh $100$ g each. On a scale with an arrow, identify all $99$-gram parts.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

The numbers $$\frac{1}{i+j-1} \,\,\,\,\,\,\, (i = 1,2,...,n; j = 1,2,...,n)$$ are written in an $n$ by $n$ table: the number $1/(i + j - 1)$ stands at the intersection of the $i$-th row and $j$-th column. Chose any $n$ squares of the table so that no two of them stand in the same row and no two of them stand in the same column. Prove that the sum of the numbers in these $n$ squares is not less than $1$. (Sergey Ivanov, St Petersburg)

Let $ABC$ be any triangle with $\angle BAC \le \angle ACB \le \angle CBA$. Let $D, E$ and $F$ be the midpoints of $BC, CA$ and $AB$, respectively, and let $\epsilon$ be a positive real number. Suppose there is an ant (represented by a point $T$ ) and two spiders (represented by points $P_1$ and $P_2$, respectively) walking on the sides $BC, CA, AB, EF, FD$ and $DE$. The ant and the spiders may vary their speeds, turn at an intersection point, stand still, or turn back at any point; moreover, they are aware of their and the others’ positions at all time. Assume that the ant’s speed does not exceed $1$ mm/s, the first spider’s speed does not exceed $\frac{\sin A}{2 \sin A+\sin B}$ mm/s, and the second spider’s speed does not exceed $\epsilon$ mm/s. Show that the spiders always have a strategy to catch the ant regardless of the starting points of the ant and the spiders. Note: the two spiders can discuss a plan before the hunt starts and after seeing all three starting points, but cannot communicate during the hunt.

There are three types of tiles: an L-shaped tile with three $1\times 1$ squares, a $2\times 2$ square, and a Z-shaped tile with four $1\times 1$ squares. We tile a $(2n-1)\times (2n-1)$ square using these tiles. Prove that there are at least $4n-1$ L-shaped tiles. I'm sorry about my poor description, but I don't know how to draw pictures...

In the fictional country of Mahishmati, there are $50$ cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city $A$, an ordered list of cities $C_1,\ldots, C_{50}$ is called an [i]antitour[/i] from $A$ if [list] [*] every city (including $A$) appears in the list exactly once, and [*] for each $k\in \{1,2,\ldots, 50\}$, it is impossible to go from $A$ to $C_k$ by a sequence of exactly $k$ (not necessarily distinct) flights. [/list] Baahubali notices that there is an antitour from $A$ for any city $A$. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city. [i]Proposed by Sutanay Bhattacharya[/i]

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Let $f$ be defined as below for integers $n \ge 0$ and $a_0, a_1, ...$ such that $\sum_{i\ge 0}a_i$ is finite: $$f(n; a_0, a_1, ...) = \begin{cases} a_{2020}, & \text{ $n = 0$} \\ \sum_{i\ge 0} a_i f(n-1;a_0,...,a_{i-1},a_i-1,a_{i+1}+1,a_{i+2},...)/ \sum_{i\ge 0}a_i & \text{$n > 0$} \end{cases}$$. Find the nearest integer to $f(2020^2; 2020, 0, 0, ...)$.

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$.

Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.

Ahuiliztli is playing around with some coins (pennies, nickels, dimes, and quarters). She keeps grabbing $k$ coins and calculating the value of her handful. After a while, she begins to notice that if $k$ is even, she more often gets even sums, and if $k$ is odd, she more often gets odd sums. Help her prove this true! Given $k$ coins chosen uniformly and at random, prove that. the probability that the parity of $k$ is the same as the parity of the $k$ coins' value is greater than the probability that the parities are different.

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods: [i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts. [i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut. Finally, stage 3 varies among the three methods as follows: [i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps. [i]Method 2:[/i] Nickolas takes the two middle size heaps. [i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice. Determine the most and the least profitable method for Nickolas.

We have 101 coins and a two-pan scale. In one weighing, we can compare the weights of two coins. What is the smallest number of weighings required in order to decide whether there exist 51 coins which all have the same weight?

In a hockey tournament, there is an odd number $n$ of teams. Each team plays exactly one match against each of the other teams. In this tournament, each team receives $2$ points for a win, $1$ point for a draw, and $0$ points for a loss. At the end of the tournament, it was observed that all the points obtained by the $n$ teams were different. For each $n$, determine the maximum possible number of draws that could have occurred in this tournament.

$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player". [b](Condition)[/b] For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$. It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.

An $n \times m$ matrix is nice if it contains every integer from $1$ to $mn$ exactly once and $1$ is the only entry which is the smallest both in its row and in its column. Prove that the number of $n \times m$ nice matrices is $(nm)!n!m!/(n+m-1)!$.

Let $N$ and $K$ be positive integers satisfying $1 \leq K \leq N$. A deck of $N$ different playing cards is shuffled by repeating the operation of reversing the order of $K$ topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than $(2N/K)^2$.

Given is a grid 11x11 with 121 cells. Four of them are colored in black, the rest are white. We have to cut a completely white rectangle (it could be a square and the rectangle must have its sides parralel to the lines of the grid), so that this rectangle has maximal possible area. What largest area of this rectangle we can guarantee? (We can cut this rectangle for every placement of the black squares)

Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds: [b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty. [b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.

Max flips $2020$ fair coins. Let the probability that there are at most $505$ heads be $p$. Estimate $-\log_2(p)$ to 5 decimal places, in the form $x.abcde$ where $x$ is a positive integer and $a, b, c, d, e$ are decimal digits.

Find the smallest positive integer $n$ with the following property: in every sequence of $n$ positive integers such that the sum of the $n$ numbers is equal to $2013$, there are some consecutive terms whose sum is equal to $31$.

A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one. A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!) Prove that a simple graph is permutationary if and only if its complement and itself are divisibility. [i]Proposed by Morteza Saghafian[/i] .