Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g.

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

Two points $J$ and $H$ lie $26$ units apart on a given plane. Let $M$ be the locus of points $T$ on this plane such that $JT^2 + HT^2 = 2020$. Then, M encloses a region on the plane with area $a$ and perimeter $p$. If $q$ and $r$ are coprime positive integers and $\frac{a}{p} = \frac{q}{r}$ , then compute $q + r$.

Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$. Find the length of the segment that connects the midpoints of the bases of the trapezium.

In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that: $ R^2\equal{}\frac{(ab\plus{}cd)(ac\plus{}bd)(ad\plus{}bc)}{16Q^2}$. Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

Determine all pairs $\{a, b\}$ of positive integers with the property that, in whatever manner you color the positive integers with two colors $A$ and $B$, there always exist two positive integers of color $A$ having their difference equal to $a$ [b]or[/b] of color $B$ having their difference equal to $b$.

A [i]palindromic table[/i] is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. \[ \begin{array}[h]{ccc} O & M & O \\ N & M & N \\ O & M & O \end{array} \] How many palindromic tables are there that use only the letters $O$ and $M$? (The table may contain only a single letter.) [i]Proposed by Evan Chen[/i]

Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that: \[ w^{2p}+x^{2p}+y^{2p}=z^{2p}. \]

Given is the non-right triangle $ABC$. $D,E$ and $F$ are the feet of the respective altitudes from $A,B$ and $C$. $P,Q$ and $R$ are the respective midpoints of the line segments $EF$, $FD$ and $DE$. $p \perp BC$ passes through $P$, $q \perp CA$ passes through $Q$ and $r \perp AB$ passes through $R$. Prove that the lines $p, q$ and $r$ pass through one point.

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

If $ f(x)\equal{}\frac{x(x\minus{}1)}{2}$, then $ f(x\plus{}2)$ equals: $ \textbf{(A)}\ f(x)\plus{}f(2) \qquad\textbf{(B)}\ (x\plus{}2)f(x) \qquad\textbf{(C)}\ x(x\plus{}2)f(x) \qquad\textbf{(D)}\ \frac{xf(x)}{x\plus{}2}\\ \textbf{(E)}\ \frac{(x\plus{}2)f(x\plus{}1)}{x}$

Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$

Three congruent circles have a common point $ O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point $ O$ are collinear.

Is it possible to fill a table $1\times n$ with pairwise distinct integers such that for any $k = 1, 2,\ldots, n$ one can find a rectangle $1\times k$ in which the sum of the numbers equals $0$ if a) $n= 11$; b) $n= 12$?

Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive.

[u]Round 1 [/u] [b]p1. [/b]Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the 12th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p2.[/b] Show that in the sequence $10017$, $100117$, $1001117$, $...$ all numbers are divisible by $53$. [b]p3.[/b] Harry and Draco have three wands: a bamboo wand, a willow wand, and a cherry wand, all of the same length. They must perform a spell wherein they take turns picking a wand and breaking it into three parts – first Harry, then Draco, then Harry again. But in order for the spell to work, Harry has to make sure it is possible to form three triangles out of the pieces of the wands, where each triangle has a piece from each wand. How should he break the wands to ensure the success of the spell? [b]p4.[/b] A $2\times 2\times 2$ cube has $4$ equal squares on each face. The squares that share a side are called neighbors (thus, each square has $4$ neighbors – see picture). Is it possible to write an integer in each square in such a way that the sum of each number with its $4$ neighbors is equal to $13$? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/8/4/0f7457f40be40398dee806d125ba26780f9d3a.png[/img] [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2 [/u] [b]p6.[/b] A red square is placed on a table. $2010$ white squares, each the same size as the red square, are then placed on the table in such a way that the red square is fully covered and the sides of every white square are parallel to the sides of the red square. Is it always possible to remove one of the white squares so the red square remains completely covered? [b]p7.[/b] A computer starts with a given positive integer to which it randomly adds either $54$ or $77$ every second and prints the resulting sum after each addition. For example, if the computer is given the number $1$, then a possible output could be: $1$, $55$, $109$, $186$, $…$ Show that after finitely many seconds the computer will print a number whose last two digits are the same. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ A$ and $ B$ be two disjoint sets in the interval $ (0,1)$ . Denoting by $ \mu$ the Lebesgue measure on the real line, let $ \mu(A)>0$ and $ \mu(B)>0$ . Let further $ n$ be a positive integer and $ \lambda \equal{}\frac1n$ . Show that there exists a subinterval $ (c,d)$ of $ (0,1)$ for which $ \mu(A\cap (c,d))\equal{}\lambda \mu(A)$ and $ \mu(B\cap (c,d))\equal{}\lambda \mu(B)$ . Show further that this is not true if $ \lambda$ is not of the form $ \frac1n$.

Two dueling wizards are at an altitude of $100$ above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by $a$ for me and by $b$ for my rival" where $a$ and $b$ are real numbers such that $0 < a < b$. Different spells have different values for $a$ and $b$. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells such that the second wizard has a guaranteed win, if the number of spells is $(a)$ finite; $(b)$ infinite?

Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime. [i]Evan o'Dorney[/i]

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

On each OMA lottery ticket there is a $9$-digit number that only uses the digits $1, 2$ and $3$ (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the $9$ figures, then they are of different colors. Bill $122222222$ is red, $222222222$ is green, what color is bill $123123123$?

If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.

Given a $3\times 3$ grid, we call the remainder of the grid an “[i]angle[/i]” when a $2\times 2$ grid is cut out from the grid. Now we place some [i]angles[/i] on a $10\times 10$ grid such that the borders of those [i]angles[/i] must lie on the grid lines or its borders, moreover there is no overlap among the [i]angles[/i]. Determine the maximal value of $k$, such that no matter how we place $k$ [i]angles[/i] on the grid, we can always place another [i]angle[/i] on the grid.

A rectangular sheet of paper is folded so that one corner lies on top of the corner diagonally opposite. The resulting shape is a pentagon whose area is $20\%$ one-sheet thick, and $80\%$ two-sheets-thick. Determine the ratio of the two sides of the original sheet of paper.