Olympiad problems

2024 ELMO Problems, 6

For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) [i]Aprameya Tripathy[/i]

2022 Novosibirsk Oral Olympiad in Geometry, 6

Tags: geometry , angle bisector

A triangle $ABC$ is given in which $\angle BAC = 40^o$. and $\angle ABC = 20^o$. Find the length of the angle bisector drawn from the vertex $C$, if it is known that the sides $AB$ and $BC$ differ by $4$ centimeters.

2019 Taiwan APMO Preliminary Test, P5

Tags: set theory , number theory , set

Find the minimum positive integer $n$ such that for any set $A$ with $n$ positive intergers has $15$ elements which sum is divisible by $15$.

2016 CHMMC (Fall), 1

Tags: number theory

Let $a_n$ be the $n$th positive integer such that when $n$ is written in base $3$, the sum of the digits of $n$ is divisible by $3$. For example, $a_1 = 5$ because $5 = 12_3$. Compute $a_{2016}$.

2009 Regional Olympiad of Mexico Northeast, 1

Tags: sequence , algebra

Consider the sequence $\{1,3,13,31,...\}$ that is obtained by following diagonally the following array of numbers in a spiral. Find the number in the $100$th position of that sequence. [img]https://cdn.artofproblemsolving.com/attachments/b/d/3531353472a748e3e0b1497a088472691f67fd.png[/img]

2021 LMT Spring, A17

Tags:

Given that the value of \[\sum_{k=1}^{2021} \frac{1}{1^2+2^2+3^2+\cdots+k^2}+\sum_{k=1}^{1010} \frac{6}{2k^2-k}+\sum_{k=1011}^{2021} \frac{24}{2k+1}\] can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Aidan Duncan[/i]

1979 IMO Longlists, 2

Tags: floor function , projective geometry , combinatorics

For a finite set $E$ of cardinality $n \geq 3$, let $f(n)$ denote the maximum number of $3$-element subsets of $E$, any two of them having exactly one common element. Calculate $f(n)$.

2012 Online Math Open Problems, 48

Tags: modular arithmetic , quadratic , gauss , geometric series

Suppose that \[\sum_{i=1}^{982} 7^{i^2}\] can be expressed in the form $983q + r$, where $q$ and $r$ are integers and $0 \leq r \leq 492$. Find $r$. [i]Author: Alex Zhu[/i]

2017 Princeton University Math Competition, A1/B3

Tags: algebra

Let $a \diamond b = ab-4(a+b)+20$. Evaluate \[1\diamond(2\diamond(3\diamond(\cdots(99\diamond100)\cdots))).\]

2019 Balkan MO Shortlist, C4

Tags: combinatorics , graph theory

A town-planner has built an isolated city whose road network consists of $2N$ roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise. Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of $N$ is this possible?

2009 Postal Coaching, 2

Tags: diophantine , diophantine equation , number theory

Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$

LMT Accuracy Rounds, 2022 S1

Tags: geometry

Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches. [img]https://cdn.artofproblemsolving.com/attachments/3/3/86f0ae7465e99d3e4bd3a816201383b98dc429.png[/img]

2025 SEEMOUS, P2

Tags: calculus , integration , taylor series

Calculate $$\lim_{n\rightarrow\infty}n\int_0^{\infty} e^{-x}\sqrt[n]{e^x - 1 -\frac{x}{1!} - \frac{x^2}{2!} - \dots -\frac{x^n}{n!}}\,dx.$$

2018 China Team Selection Test, 3

Tags: inequalities , algebra

Prove that there exists a constant $C>0$ such that $$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$ holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$

2014 BMT Spring, 1

Tags: number theory

A [i]festive [/i] number is a four-digit integer containing one of each of the digits $0, 1, 2$, and $4$ in its decimal representation. How many festive numbers are there?

1989 IMO, 6

Tags: counting , injective function , combinatorial inequality , combinatorics

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.

2016 Iran Team Selection Test, 4

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

1987 Traian Lălescu, 1.3

Tags: geometry , 3d geometry , tetrahedron

Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.

PEN N Problems, 13

Tags: algebra , polynomial , arithmetic sequence , more sequences

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

1999 IMC, 1

Tags: linear algebra , matrix , algebra , polynomial , complex number

a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$. b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.

2003 Romania National Olympiad, 4

Tags: function , combinatorial geometry

Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral. [i]Dinu Şerbănescu[/i]

2013 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , integer

If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer

1989 China Team Selection Test, 3

Tags: combinatorics

$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.

May Olympiad L1 - geometry, 2021.1

Tags: geometry

In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.

2009 Miklós Schweitzer, 3

Tags: search , limit , set theory , combinatorics

Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then \[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]