A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$. Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.

On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.

Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Several square-shaped papers are situated on a table such that every side of the paper is positioned parallel to the sides of the table. Each paper has a colour, and there are $n$ different coloured papers. It is known that for every $n$ papers with distinct colors, we can always find an overlapping pair of papers. Prove that, using $2n- 2$ nails, it is possible to hammer all the squares of a certain colour to the table.

Seven identical-looking coins are given, of which four are real and three are counterfeit. The three counterfeit coins have equal weight, and the four real coins have equal weight. It is known that a counterfeit coin is lighter than a real one. In one weighing, one can select two sets of coins and check which set has a smaller total weight, or if they are of equal weight. How many weightings are needed to identify one counterfeit coin?

In the plane are given $1997$ points. Show that among the pairwise distances between these points, there are at least $32$ different values.

For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?

We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-rectangles, one form one rectangle of size $1993 \times 2000$ (figures don’t overlap each other). Let $s$ denote the sum of numbers of squares and $S$-rectangles used in such tiling. Find the maximal value of $s$.

Suppose you have identical coins distributed in several piles with one or more coins in each pile. An action consists of taking two piles, which have an even total of coins among them, and redistribute their coins in two piles so that they end up with the same number of coins. A distribution is [i]levelable[/i] if it is possible, by means of 0 or more operations, to end up with all the piles having the same number of coins. Determine all positive integers $n$ such that, for all positive integers $k$, any distribution of $nk$ coins in $n$ piles is levelable.

In a pile you have 100 stones. A partition of the pile in $ k$ piles is [i]good[/i] if: 1) the small piles have different numbers of stones; 2) for any partition of one of the small piles in 2 smaller piles, among the $ k \plus{} 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone). Find the maximum and minimal values of $ k$ for which this is possible.

A numismatist Fred has some coins. A diameter of any coin is no more than $10$ cm. All the coins are contained in a one-layer box of dimensions $30$ cm by $70$ cm. He is presented with a new coin. Its diameter is $25$ cm. Prove that it is possible to put all the coins in a one-layer box of dimensions $55$ cm by $55$ cm. (Fedja Nazarov, St Petersburg)

Let $p \geqslant 5$ be a prime and $S = \left\{ 1, 2, \ldots, p \right\}$. Define $r(x,y)$ as follows: \[ r(x,y) = \begin{cases} y - x & y \geqslant x \\ y - x + p & y < x \end{cases}.\] For a nonempty proper subset $A$ of $S$, let $$f(A) = \sum_{x \in A} \sum_{y \in A} \left( r(x,y) \right)^2.$$A [i]good[/i] subset of $S$ is a nonempty proper subset $A$ satisfying that for all subsets $B \subseteq S$ of the same size as $A$, $f(B) \geqslant f(A)$. Find the largest integer $L$ such that there exists distinct good subsets $A_1 \subseteq A_2 \subseteq \ldots \subseteq A_L$. [i]Proposed by Bin Wang[/i]

We removed the middle square of $2 \times 2$ from the $8 \times 8$ board. a) How many checkers can be placed on the remaining $60$ boxes so that there are no two not jeopardize? b) How many at least checkers can be placed on the board so that they are at risk all $60$ squares? (A lady is threatening the box she stands on, as well as any box she can get to in one move without going over any of the four removed boxes.)

Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)

For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$

A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?

Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.

Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

One-round chess tournament involves $ 10 $ players from two countries. For a victory, one point is given, for a draw - half a point, for defeat - zero. All players scored a different number of points. Prove that one of the chess players scored in meetings with his countrymen less points, than meeting with players from another country.

Prove that in the set $ \{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $ A_i, \{i \equal{} 1,2, \ldots, 117\}$ such that [b]i.)[/b] each $ A_i$ contains 17 elements [b]ii.)[/b] the sum of all the elements in each $ A_i$ is the same.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$. [i]Proposed by Christopher Cheng[/i]