Olympiad problems

2006 AMC 8, 16

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read? $ \textbf{(A)}\ 6400 \qquad \textbf{(B)}\ 6600 \qquad \textbf{(C)}\ 6800 \qquad \textbf{(D)}\ 7000 \qquad \textbf{(E)}\ 7200$

1973 Bundeswettbewerb Mathematik, 3

Tags:

Given $n$ digits $a_{1}, a_{2},..., a_{n}$ in that order. Does there exist a positive integer such that the first $n$ decimal digits after the dot of that number's square root are exactly those given digits¿ Give reason for your answer.

2013 AMC 12/AHSME, 20

Tags: symmetry , pigeonhole principle , inequalities

Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \leq 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$? $ \textbf{(A)} \ 810 \qquad \textbf{(B)} \ 855 \qquad \textbf{(C)} \ 900 \qquad \textbf{(D)} \ 950 \qquad \textbf{(E)} \ 988$

2014 Taiwan TST Round 1, 3

Tags: rectangle , incenter , geometric transformation , reflection , geometry

Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.

1996 Israel National Olympiad, 2

Tags: polynomial , algebra , functional equation , functional

Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

2023 ELMO Shortlist, A4

Tags: algebra

Let $f:\mathbb R\to\mathbb R$ be a function such that for all real numbers $x\neq1$, \[f(x-f(x))+f(x)=\frac{x^2-x+1}{x-1}.\] Find all possible values of $f(2023)$. [i]Proposed by Linus Tang[/i]

2023 Benelux, 2

Tags: combinatorics

Determine all integers $k\geqslant 1$ with the following property: given $k$ different colours, if each integer is coloured in one of these $k$ colours, then there must exist integers $a_1<a_2<\cdots<a_{2023}$ of the same colour such that the differences $a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}$ are all powers of $2$.

2013 AMC 12/AHSME, 12

Tags: trigonometry , arithmetic sequence , trig identities , law of cosines

The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$? $ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 38\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 44$

2005 Purple Comet Problems, 22

Tags:

Let $d_k$ be the greatest odd divisor of $k$ for $k = 1, 2, 3, \ldots$. Find $d_1 + d_2 + d_3 + \ldots + d_{1024}$.

2022 BMT, 17

Tags: combinatorics

Midori and Momoi are arguing over chores. Each of $5$ chores may either be done by Midori, done by Momoi, or put off for tomorrow. Today, each of them must complete at least one chore, and more than half of the chores must be completed. How many ways can they assign chores for today? (The order in which chores are completed does not matter.)

2012 Estonia Team Selection Test, 2

Tags: number theory , sequence

For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold: (1) $a_i = a_{i+n}$ for any $i$, (2) $a_i$ is not divisible by $n$ for any $i$, (3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$. For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?

1958 AMC 12/AHSME, 35

Tags: analytic geometry , geometry

A triangle is formed by joining three points whose coordinates are integers. If the $ x$-coordinate and the $ y$-coordinate each have a value of $ 1$, then the area of the triangle, in square units: $ \textbf{(A)}\ \text{must be an integer}\qquad \textbf{(B)}\ \text{may be irrational}\qquad \textbf{(C)}\ \text{must be irrational}\qquad \textbf{(D)}\ \text{must be rational}\qquad \\ \textbf{(E)}\ \text{will be an integer only if the triangle is equilateral.}$

2008 Bulgaria National Olympiad, 1

Tags: circumcircle , trigonometry , angle bisector , geometry

Let $ ABC$ be an acute triangle and $ CL$ be the angle bisector of $ \angle ACB$. The point $ P$ lies on the segment $CL$ such that $ \angle APB\equal{}\pi\minus{}\frac{_1}{^2}\angle ACB$. Let $ k_1$ and $ k_2$ be the circumcircles of the triangles $ APC$ and $ BPC$. $ BP\cap k_1\equal{}Q, AP\cap k_2\equal{}R$. The tangents to $ k_1$ at $ Q$ and $ k_2$ at $ B$ intersect at $ S$ and the tangents to $ k_1$ at $ A$ and $ k_2$ at $ R$ intersect at $ T$. Prove that $ AS\equal{}BT.$

2020 CMIMC Combinatorics & Computer Science, 10

Tags: combinatorics , computer science

Define a string to be doubly palindromic if it can be split into two (non-empty) parts that are read the same both backwards and forwards. For example hannahhuh is doubly palindromic as it can be split into hannah and huh. How many doubly palindromic strings of length 9 using only the letters $\{a, b, c, d\}$ are there?

2016 Iran Team Selection Test, 6

Tags: function , number theory , greatest common divisor

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

2025 Ukraine National Mathematical Olympiad, 11.5

Tags: algebra , polynomial

Initially, two constant polynomials are written on the board: $0$ and $1$. At each step, it is allowed to add $1$ to one of the polynomials and to multiply another one by the polynomial $45x + 2025$. Can the polynomials become equal at some point? [i]Proposed by Oleksii Masalitin[/i]

Estonia Open Senior - geometry, 2003.2.4

Tags: ratio , geometry , cevian

Consider the points $D, E$ and $F$ on the respective sides $BC, CA$ and $AB$ of the triangle $ABC$ in a way that the segments $AD, BE$ and $CF$ have a common point $P$. Let $\frac{|AP|}{|PD|}= x,$ $\frac{|BP|}{|PE|}= y$ and $\frac{|CP|}{|PF|}= z$. Prove that $xyz - (x + y + z) = 2$.

PEN M Problems, 26

Tags: modular arithmetic , function , inequalities , floor function , recursive sequences

Let $p$ be an odd prime $p$ such that $2h \neq 1 \; \pmod{p}$ for all $h \in \mathbb{N}$ with $h< p-1$, and let $a$ be an even integer with $a \in] \tfrac{p}{2}, p [$. The sequence $\{a_n\}_{n \ge 0}$ is defined by $a_{0}=a$, $a_{n+1}=p -b_{n}$ \; $(n \ge 0)$, where $b_{n}$ is the greatest odd divisor of $a_n$. Show that the sequence $\{a_n\}_{n \ge 0}$ is periodic and find its minimal (positive) period.

2017 Israel National Olympiad, 1

Tags: ratio , geometry , area

[list=a] [*] In the right picture there is a square with four congruent circles inside it. Each circle is tangent to two others, and to two of the edges of the square. Evaluate the ratio between the blue part and white part of the square's area. [*] In the left picture there is a regular hexagon with six congruent circles inside it. Each circle is tangent to two others, and to one of the edges on the hexagon in its midpoint. Evaluate the ratio between the blue part and white part of the hexagon's area. [/list] [img]https://i.imgur.com/fAuxoc9.png[/img]

2015 Harvard-MIT Mathematics Tournament, 6

Tags:

In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees.

2020 Czech-Austrian-Polish-Slovak Match, 2

Tags: combinatorics , algebra

Given a positive integer $n$, we say that a real number $x$ is $n$-good if there exist $n$ positive integers $a_1,...,a_n$ such that $$x=\frac{1}{a_1}+...+\frac{1}{a_n}.$$ Find all positive integers $k$ for which the following assertion is true: if $a,b$ are real numbers such that the closed interval $[a,b]$ contains infinitely many $2020$-good numbers, then the interval $[a,b]$ contains at least one $k$-good number. (Josef Tkadlec, Czech Republic)

2013 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry

Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.

2021 HMNT, 8

Tags: combinatorics , geometry

Let $n$ be the answer to this problem. Find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.

2013 Irish Math Olympiad, 1

Tags: exponent , number theory

Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.

Indonesia MO Shortlist - geometry, g1

Tags: geometry , geometric transformation , reflection

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .