Olympiad problems

2009 JBMO Shortlist, 2

Tags: number theory

A group of $n > 1$ pirates of different age owned total of $2009$ coins. Initially each pirate (except the youngest one) had one coin more than the next younger. a) Find all possible values of $n$. b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If $n = 7$, find the largest possible number of coins a pirate can have after several days.

LMT Guts Rounds, 2020 F20

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Cyclic quadrilateral $ABCD$ has $AC=AD=5, CD=6,$ and $AB=BC.$ If the length of $AB$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,c$ are relatively prime positive integers and $b$ is square-fre,e evaluate $a+b+c.$ [i]Proposed by Ada Tsui[/i]

2013 Hanoi Open Mathematics Competitions, 11

Tags: polynomial , minimum , maximum

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

2013 NIMO Problems, 6

Tags: circumcircle , function , trigonometry , trig identities , law of cosines , geometry

Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$. [i]Proposed by Yang Liu[/i]

2003 Germany Team Selection Test, 2

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Given a triangle $ABC$ and a point $M$ such that the lines $MA,MB,MC$ intersect the lines $BC,CA,AB$ in this order in points $D,E$ and $F,$ respectively. Prove that there are numbers $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}$ such that: \[\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.\]

1971 Miklós Schweitzer, 5

Tags: real analysis

Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\] [i]L. Leindler[/i]

2000 Moldova National Olympiad, Problem 5

Tags: function , induction , algebra , logarithm

Let $ p$ be a positive integer. Define the function $ f: \mathbb{N}\to\mathbb{N}$ by $ f(n)\equal{}a_1^p\plus{}a_2^p\plus{}\cdots\plus{}a_m^p$, where $ a_1, a_2,\ldots, a_m$ are the decimal digits of $ n$ ($ n\equal{}\overline{a_1a_2\ldots a_m}$). Prove that every sequence $ (b_k)^\infty_{k\equal{}0}$ of positive integer that satisfy $ b_{k\plus{}1}\equal{}f(b_k)$ for all $ k\in\mathbb{N}$, has a finite number of distinct terms. $ \mathbb{N}\equal{}\{1,2,3\ldots\}$

2000 Slovenia National Olympiad, Problem 4

Tags: analytic geometry , geometry

All vertices of a convex $n$-gon ($n\ge3$) in the plane have integer coordinates. Show that its area is at least $\frac{n-2}2$.

2011 Ukraine Team Selection Test, 9

Tags: geometry , circumcircle , equal angles , cyclic quadrilateral , tangent , tangent circle

Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.

2009 Jozsef Wildt International Math Competition, W. 2

Tags: algebra

Find the area of the set $A = \{(x, y)\ |\ 1 \leq x \leq e,\ 0 \leq y \leq f (x)\}$, where \begin{tabular}{ c| c c c c |} &1 & 1& 1 & 1\\ $f(x)$=& $\ln x$ & 2$\ln x$ & 3$\ln x$ & 4$\ln x$ \\ &${(\ln x)}^2$ & $4{(\ln x)}^2 $& $9{(\ln x)}^2 $& $16{(\ln x)}^2$\\ &${(\ln x)}^3$ & $8{(\ln x)}^3$ &$ 27{(\ln x)}^3$ &$ 64{(\ln x)}^3$ \end{tabular}

2018 Korea National Olympiad, 2

Tags: combinatorics , counting

For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+y+2z+3w=n-1$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that (i). $a+b+c+d=n$. (ii). $a \ge b$, $c \ge d$, $a \ge d$. (iii). $b < c$. Prove that for all $n$, $p(n) = q(n)$.

2016 Canadian Mathematical Olympiad Qualification, 1

Tags: number theory

(a) Find all positive integers $n$ such that $11|(3^n + 4^n)$. (b) Find all positive integers $n$ such that $31|(4^n + 7^n + 20^n)$.

2022 Malaysian IMO Team Selection Test, 5

Tags: number theory

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

2022 Putnam, A2

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Let $n$ be an integer with $n\geq 2.$ Over all real polynomials $p(x)$ of degree $n,$ what is the largest possible number of negative coefficients of $p(x)^2?$

2020 Mediterranean Mathematics Olympiad, 3

Tags: algebra , inequalities

Prove that all postive real numbers $a,b,c$ with $a+b+c=4$ satisfy the inequality $$\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}$$

1988 Romania Team Selection Test, 5

Tags: geometry , rectangle , combinatorics

The cells of a $11\times 11$ chess-board are colored in 3 colors. Prove that there exists on the board a $m\times n$ rectangle such that the four cells interior to the rectangle and containing the four vertices of the rectangle have the same color. [i]Ioan Tomescu[/i]

1966 IMO Shortlist, 14

Tags: combinatorics , combinatorial geometry , counting , circles , arrangement

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

2019 India PRMO, 17

Tags: algebra

Let $a,b,c$ be distinct positive integers such that $b+c-a$, $c+a-b$ and $a+b-c$ are all perfect squares. What is the largest possible value of $a+b+c$ smaller than $100$ ?

2001 Moldova Team Selection Test, 12

Tags: number theory

Let $n{}$ $(n\geq 1)$ be an integer and a set $A=\{1,2,\ldots,n\}$. The set $A{}$ is $k-partitionable$ if it can be partitioned in $k{}$ disjoint sets with the same sum of elements. Show that $A{}$ is $k-partitionable$ if and only if $2k$ divides $n(n+1)$ and $2k\leq n+1$.

2022 Princeton University Math Competition, A3 / B5

Tags: combinatorics

Randy has a deck of $29$ distinct cards. He chooses one of the $29!$ permutations of the deck and then repeatedly rearranges the deck using that permutation until the deck returns to its original order for the first time. What is the maximum number of times Randy may need to rearrange the deck?

2017 Lusophon Mathematical Olympiad, 6

Tags: geometry , circumcircle

Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that $\overline{AF}$=$\overline{DF}$ and $\overline{BE}$=$\overline{DE}$. Show that the circumcenter of ABC lies on the circumcircle of CEF.

2017 Tournament Of Towns, 5

Tags: combinatorics , number theory

There is a set of control weights, each of them weighs a non-integer number of grams. Any integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control weights are on one balance pan, and the measured weight on the other pan).What is the least possible number of the control weights? [i](Alexandr Shapovalov)[/i]

1994 Tournament Of Towns, (399) 1

Tags: geometry , geometric construction , quadrilateral construction , construction

Construct a convex quadrilateral given the lengths of all its sides and the length of the segment between the midpoints of its diagonals. (Folklore)

1962 IMO, 4

Tags: trigonometric equation , trigonometry , algebra , equation

Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

1989 Dutch Mathematical Olympiad, 4

Tags: geometry , fixed , 3d geometry , pyramid

Given is a regular $n$-sided pyramid with top $T$ and base $A_1A_2A_3... A_n$. The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$, the plane $TA_2A_3$ at $C_2$, and so on, and finally the plane $TA_nA_1$ at $C_n$. Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$'s.