Olympiad problems

2022 AMC 12/AHSME, 23

Tags: AMC 12 , AMC , number theory , relatively prime , 2022 AMC , 2022 AMC 12A

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n}.\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3,\cdots, n$. For how many integers $n$ with $1 \le n \le 22$ is $k_n<L_n$? $\textbf{(A)} ~0 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~10 $

2017 Bosnia Herzegovina Team Selection Test, 2

Tags: number theory , IMO Shortlist , function , Divisibility , Hi

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2024 Canadian Open Math Challenge, B3

Tags: Comc

Let $a,b,c,d$ be four [b]distinct [/b]integers such that: $$\text{min}(a,b)=2$$ $$\text{min}(b,c)=0$$ $$\text{max}(a,c)=2$$ $$\text{max}(c,d)=4$$ Here $\text{min}(a,b)$ and $\text{max}(a,b)$ denote respectively the minimum and the maximum of two numbers $a$ and $b$. Determine the fifth smallest possible value for $a+b+c+d$

1955 Poland - Second Round, 1

Tags: algebra

Calculate the sum $ x^4 + y^4 + z^4 $ knowing that $ x + y + z = 0 $ and $ x^2 + y^2 + z^2 = a $, where $ a $ is a given positive number.

2002 China Team Selection Test, 2

Tags: geometry , incenter , geometry unsolved

$ \odot O_1$ and $ \odot O_2$ meet at points $ P$ and $ Q$. The circle through $ P$, $ O_1$ and $ O_2$ meets $ \odot O_1$ and $ \odot O_2$ at points $ A$ and $ B$. Prove that the distance from $ Q$ to the lines $ PA$, $ PB$ and $ AB$ are equal. (Prove the following three cases: $ O_1$ and $ O_2$ are in the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ and $ O_2$ are out of the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ is in the common space of $ \odot O_1$ and $ \odot O_2$, $ O_2$ is out of the common space of $ \odot O_1$ and $ \odot O_2$.

2012 Belarus Team Selection Test, 3

Tags: geometry , circumcircle , trigonometry , geometric transformation , IMO Shortlist

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

1960 IMO, 6

Tags: geometry , 3D geometry , sphere , trigonometry , inradius , IMO , IMO 1960

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

1959 Miklós Schweitzer, 1

Tags: number theory , prime numbers , college contests , real analysis

[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ [b](N.17)[/b]

1965 AMC 12/AHSME, 23

Tags: inequalities , function

If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, the smallest value we can use for $ N$ is: $ \textbf{(A)}\ .0301 \qquad \textbf{(B)}\ .0349 \qquad \textbf{(C)}\ .0399 \qquad \textbf{(D)}\ .0401 \qquad \textbf{(E)}\ .0499 \qquad$

2023 Balkan MO Shortlist, C3

Tags: combinatorics

In a given community of people, each person has at least two friends within the community. Whenever some people from this community sit on a round table such that each adjacent pair of people are friends, it happens that no non-adjacent pair of people are friends. Prove that there exist two people in this community such that each has exactly two friends and they have at least one common friend.

2010 HMNT, 8

Tags: geometry

Two circles with radius one are drawn in the coordinate plane, one with center $(0,1)$ and the other with center $(2, y)$, for some real number y between $0$ and $1$. A third circle is drawn so as to be tangent to both of the other two circles as well as the $x$ axis. What is the smallest possible radius for this third circle?

2022 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , divisible

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

1993 All-Russian Olympiad, 1

Tags: number theory proposed , number theory

For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?

2015 Oral Moscow Geometry Olympiad, 3

Tags: areas , trapezoid , geometry

$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.

2025 Turkey Team Selection Test, 8

Tags: analysis , Sequence , TST , Turkey , algebra

A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer $s$ is given. Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$ \[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\] Then for every integer $n\geq s,$ the condition \[a_{n+1}=\max_{0\leq k<n}(f_n(k))\] is satisfied. Prove that this sequence must be eventually constant.

2017 Putnam, A1

Tags: Putnam , Putnam 2017

Let $S$ be the smallest set of positive integers such that a) $2$ is in $S,$ b) $n$ is in $S$ whenever $n^2$ is in $S,$ and c) $(n+5)^2$ is in $S$ whenever $n$ is in $S.$ Which positive integers are not in $S?$ (The set $S$ is ``smallest" in the sense that $S$ is contained in any other such set.)

2022 Harvard-MIT Mathematics Tournament, 6

Tags: algebra , complex numbers , polynomial

Let $P(x) = x^4 + ax^3 + bx^2 + x$ be a polynomial with four distinct roots that lie on a circle in the complex plane. Prove that $ab\ne 9$.

1994 IberoAmerican, 2

Tags: geometry , rectangle , inradius , incenter , ratio , trigonometry , perimeter

Let $ ABCD$ a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on $ AB$, that is tangent to the other three sides of the cuadrilateral. (i) Show that $ AB \equal{} AD \plus{} BC$. (ii) Calculate, in term of $ x \equal{} AB$ and $ y \equal{} CD$, the maximal area that can be reached for such quadrilateral.

2024 Belarusian National Olympiad, 10.2

Tags: combinatorics

Some vertices of a regular $2024$-gon are marked such that for any regural polygon, all of whose vertices are vertices of the $2024$-gon, at least one of his vertices is marked. Find the minimal possible number of marked vertices [i]A. Voidelevich[/i]

2008 IMO Shortlist, 3

Tags: greatest common divisor , number theory , Sequence , IMO Shortlist

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

2016 Turkey EGMO TST, 6

Tags: number theory , squarefree

Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

1972 AMC 12/AHSME, 15

Tags: AMC

A contractor estimated that one of his two bricklayers would take $9$ hours to build a certain wall and the other $10$ hours. However, he knew from experience that when they worked together, their combined output fell by $10$ bricks per hour. Being in a hurry, he put both men on the job and found that it took exactly 5 hours to build the wall. The number of bricks in the wall was $\textbf{(A) }500\qquad\textbf{(B) }550\qquad\textbf{(C) }900\qquad\textbf{(D) }950\qquad \textbf{(E) }960$

2008 AMC 10, 8

Tags: AMC

Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars? $ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$

2014 IFYM, Sozopol, 2

Tags: number theory , Perfect Square

Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?

1966 Miklós Schweitzer, 2

Tags: advanced fields , advanced fields unsolved

Characterize those configurations of $ n$ coplanar straight lines for which the sum of angles between all pairs of lines is maximum. [i]L.Fejes-Toth, A. Heppes[/i]