Olympiad problems

2021 Romania Team Selection Test, 3

The external bisectors of the angles of the convex quadrilateral $ABCD$ intersect each other in $E,F,G$ and $H$ such that $A\in EH, \ B\in EF, \ C\in FG, \ D\in GH$. We know that the perpendiculars from $E$ to $AB$, from $F$ to $BC$ and from $G$ to $CD$ are concurrent. Prove that $ABCD$ is cyclic.

2006 VTRMC, Problem 6

Tags: triangle , geometry

In the diagram below, $BP$ bisects $\angle ABC$, $CP$ bisects $\angle BCA$, and $PQ$ is perpendicular to $BC$. If $BQ\cdot QC=2PQ^2$, prove that $AB+AC=3BC$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC8zL2IwZjNjMDAxNWEwMTc1ZGNjMTkwZmZlZmJlMGRlOGRhYjk4NzczLnBuZw==&rn=VlRSTUMgMjAwNi5wbmc=[/img]

2022 Korea National Olympiad, 8

Tags: prime number , number theory

$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation. $$p^2 x^4-6px^2+1=y^2$$

2006 AMC 12/AHSME, 5

Tags:

John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? $ \textbf{(A) } 30 \qquad \textbf{(B) } 50 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 120$

2010 National Chemistry Olympiad, 8

Tags:

Which compound contains the highest percentage of nitrogen by mass? $ \textbf{(A)} \text{NH}_2\text{OH} (M=33.0) \qquad\textbf{(B)}\text{NH}_4\text{NO}_2 (M=64.1)\qquad$ $\textbf{(C)}\text{N}_2\text{O}_3 (M=76.0)\qquad\textbf{(D)}\text{NH}_4\text{NH}_2\text{CO}_2 (M=78.1)\qquad $

2001 Baltic Way, 10

Tags: geometry , circumcircle

In a triangle $ABC$, the bisector of $\angle BAC$ meets the side $BC$ at the point $D$. Knowing that $|BD|\cdot |CD|=|AD|^2$ and $\angle ADB=45^{\circ}$, determine the angles of triangle $ABC$.

1990 IMO Longlists, 80

Tags: algebra , induction , function

Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

2018 ELMO Problems, 1

Tags: combinatorics

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

2003 Mexico National Olympiad, 4

Tags: geometry

The quadrilateral $ABCD$ has $AB$ parallel to $CD$. $P$ is on the side $AB$ and $Q$ on the side $CD$ such that $\frac{AP}{PB}= \frac{DQ}{CQ}$. M is the intersection of $AQ$ and $DP$, and $N$ is the intersection of $PC$ and $QB$. Find $MN$ in terms of $AB$ and $CD$.

2001 IMC, 5

Tags: function , functional equation

Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that \[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]

2004 IMO Shortlist, 6

Tags: extremal combinatorics , linear algebra , combinatorics , matrix

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

2010 Iran MO (3rd Round), 1

Tags: combinatorics

suppose that $\mathcal F\subseteq X^{(k)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B,C$, at most one of $A\cap B$,$B\cap C$ and $C\cap A$ is $\phi$. for $k\le \frac{n}{2}$ prove that: a) $|\mathcal F|\le max(1,4-\frac{n}{k})\times \dbinom{n-1}{k-1}$.(15 points) b) find all cases of equality in a) for $k\le \frac{n}{3}$.(5 points)

2004 Romania Team Selection Test, 18

Tags: quadratic , number theory , modular arithmetic

Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by \[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \] where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$). a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$. [i]Sugested by Calin Popescu.[/i]

2016 BMT Spring, 1

Tags: algebra

Define an such that $a_1 =\sqrt3$ and for all integers $i$, $a_{i+1} = a^2_i - 2$. What is $a_{2016}$?

2017 NMTC Junior, 3

Tags: geometry

$ADC$ and $ABC$ are triangles such that $AD=DC$ and $AC=AB$. If $\angle CAB=20^{\circ}$ and $\angle ADC =100^{\circ}$, without using Trigonometry, prove that $AB=BC+CD$.

2005 Abels Math Contest (Norwegian MO), 1b

Tags: geometry , 3d geometry , even , volume , integer , pyramid

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.

2023 Korea - Final Round, 4

Tags: easy , number theory

Find all positive integers $n$ satisfying the following. $$2^n-1 \text{ doesn't have a prime factor larger than } 7$$

2014 Indonesia MO, 4

Tags: number theory , modular arithmetic

A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

2019 Singapore Senior Math Olympiad, 3

Tags: least common multiple , number theory

Let $a_1,a_2,\cdots,a_{2000}$ be distinct positive integers such that $1 \leq a_1 < a_2 < \cdots < a_{2000} < 4000$ such that the LCM (least common multiple) of any two of them is $\geq 4000$. Show that $a_1 \geq 1334$

1986 Tournament Of Towns, (128) 3

Tags: cover , triangle , combinatorics , combinatorial geometry

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

2010 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: frac , algebra

It is given positive real number $a$ such that: $$\left\{\frac{1}{a}\right\}=\{a^2\}$$ $$ 2<a^2<3$$ Find the value of $$a^{12}-\frac{144}{a}$$

2008 ISI B.Math Entrance Exam, 6

Tags: algebra

Let $\dbinom{n}{k}$ denote the binomial coefficient $\frac{n!}{k!(n-k)!}$ , and $F_m$ be the $m^{th}$ Fibonacci number given by $F_1=F_2=1$ and $F_{m+2}=F_m+F_{m+1}$ for all $m\geq 1$. Show that $\sum \dbinom{n}{k}=F_{m+1}$ for all $m\geq 1$ . Here the above sum is over all pairs of integers $n\geq k\geq 0$ with $n+k=m$ .

2013 Israel National Olympiad, 3

Tags: arctangent , algebra , polynomial

Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.

2018 CMIMC Number Theory, 7

Tags: number theory

For each $q\in\mathbb Q$, let $\pi(q)$ denote the period of the repeating base-$16$ expansion of $q$, with the convention of $\pi(q)=0$ if $q$ has a terminating base-$16$ expansion. Find the maximum value among \[\pi\left(\frac11\right),~\pi\left(\frac12\right),~\dots,~\pi\left(\frac1{70}\right).\]

2022 Saudi Arabia BMO + EGMO TST, 1.1

Tags: perfect power , number theory

Find all positive integers $k$ such that the product of the first $k$ primes increased by $1$ is a power of an integer (with an exponent greater than $1$).