Olympiad problems

2018 IMO Shortlist, N6

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

2022 Purple Comet Problems, 12

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A rectangle with width $30$ inches has the property that all points in the rectangle are within $12$ inches of at least one of the diagonals of the rectangle. Find the maximum possible length for the rectangle in inches.

2023 Francophone Mathematical Olympiad, 4

Tags: number theory , divide , consecutive , product

Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?

2024 Harvard-MIT Mathematics Tournament, 4

Tags: algebra

Let $f(x)$ be a quotient of two quadratic polynomials. Given that $f(n) = n^3$ for all $n \in \{1, 2, 3, 4, 5\}$, compute $f(0)$.

2013 IFYM, Sozopol, 3

Tags: induction , relatively prime , number theory

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

2015 Korea - Final Round, 4

Tags: circumcircle , geometry

$\triangle ABC$ is an acute triangle and its orthocenter is $H$. The circumcircle of $\triangle ABH$ intersects line $BC$ at $D$. Lines $DH$ and $AC$ meets at $P$, and the circumcenter of $\triangle ADP$ is $Q$. Prove that the circumcenter of $\triangle ABH$ lies on the circumcircle of $\triangle BDQ$.

2020 CHKMO, 4

Tags: combinatorics , graph theory

There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$, there is either a one way road from $A$ to $B$, or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$. Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country.

2020 Online Math Open Problems, 9

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A magician has a hat that contains $a$ white rabbits and $b$ black rabbits. The magician repeatedly draws pairs of rabbits chosen at random from the hat, without replacement. Call a pair of rabbits \emph{checkered} if it consists of one white rabbit and one black rabbit. Given that the magician eventually draws out all the rabbits without ever drawing out an unpaired rabbit and that the expected value of the number of checkered pairs that the magician draws is $2020$, compute the number of possible pairs $(a,b)$. [i]Proposed by Ankit Bisain[/i]

2024 Chile TST IMO, 2

Tags: number theory

Find all natural numbers that have a multiple consisting only of the digit 9.

KoMaL A Problems 2020/2021, A. 802

Tags: combinatorics , geometry

Let $P$ be a given regular $100$-gon. Prove that if we take the union of two polygons that are congruent to $P,$ the ratio of the perimeter and area of the resulting shape cannot be more than the ratio of the perimeter and area of $P.$

2019 Philippine TST, 3

Tags: number theory , divisor , bounding , infinite sequence

Let $a_1, a_2, a_3,\ldots$ be an infinite sequence of positive integers such that $a_2 \ne 2a_1$, and for all positive integers $m$ and $n$, the sum $m + n$ is a divisor of $a_m + a_n$. Prove that there exists an integer $M$ such that for all $n > M$, we have $a_n \ge n^3$.

2008 Singapore Senior Math Olympiad, 2

Tags: number theory , perfect square

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

2020/2021 Tournament of Towns, P5

Tags: number theory

Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others? [i]Mikhail Evdokimov[/i]

2012 Balkan MO Shortlist, A4

Tags: geometry , minimum value , maximum value

Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$

2005 Moldova Team Selection Test, 4

Tags: function , number theory

Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.

Kvant 2025, M2831

Tags: geometry , parabola , conic

Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$). [i]From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov[/i]

2020 IMO Shortlist, A4

Tags: inequalities , algebra , mount inquality

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

2007 IMO Shortlist, 1

Tags: combinatorics , sequence , inequality system , induction

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

2021 Stanford Mathematics Tournament, 3

Tags: geometry

In quadrilateral $ABCD$, $CD = 14$, $\angle BAD = 105^o$, $\angle ACD = 35^o$, and $\angle ACB = 40^o$. Let the midpoint of $CD$ be $M$. Points $P$ and $Q$ lie on $\overrightarrow{AM}$ and $\overrightarrow{BM}$, respectively, such that $\angle AP B = 40^o$ and $\angle AQB = 40^o$ . $P B$ intersects $CD$ at point $R$ and $QA$ intersects $CD$ at point $S$. If $CR = 2$, what is the length of $SM$?

2023 IFYM, Sozopol, 1

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Do there exist distinct natural numbers $x, y, z, t$, all greater than or equal to $2$, such that $x \geq y + 2$, $z \geq t + 2$, and \[ \binom{x}{y} = \binom{z}{t}? \] [i](For natural numbers $n$ and $k$ with $n \geq k$, we define $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.)[/i]

1999 Harvard-MIT Mathematics Tournament, 8

Tags: algebra

Find all the roots of $(x^2 + 3x + 2)(x^2 - 7x + 12)(x^2- 2x -1) + 24 = 0$.

2004 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

Let $ACE$ be a triangle with a point $B$ on segment $AC$ and a point $D$ on segment $CE$ such that $BD$ is parallel to $AE$. A point $Y$ is chosen on segment $AE$, and segment $CY$ is drawn. Let $X$ be the intersection of $CY$ and $BD$. If $CX = 5$, $XY = 3$, what is the ratio of the area of trapezoid $ABDE$ to the area of triangle $BCD$? [img]https://cdn.artofproblemsolving.com/attachments/9/2/d6c723e54dbc4c88c12aa6a6ee91ae9e3ea581.png[/img]

2010 Stars Of Mathematics, 2

Tags: geometric transformation , reflection , perpendicular bisector , geometry

Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$. 1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$; 2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$. (Dan Schwarz)

2006 National Olympiad First Round, 20

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The integer $k$ is a [i]good number[/i], if we can divide a square into $k$ squares. How many good numbers not greater than $2006$ are there? $ \textbf{(A)}\ 1003 \qquad\textbf{(B)}\ 1026 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 2003 \qquad\textbf{(E)}\ 2004 $

2021 Thailand TSTST, 3

Tags: polynomial , algebra

Let $m, n$ be positive integers. Show that the polynomial $$f(x)=x^m(x^2-100)^n-11$$ cannot be expressed as a product of two non-constant polynomials with integral coeﬀicients.