Olympiad problems

1997 Brazil Team Selection Test, Problem 1

Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.

2014 AMC 8, 3

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Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book? $\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$

2002 Moldova National Olympiad, 1

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We are given three nuggets of weights $ 1$ kg, $ 2$ kg and $ 3$ kg, containing different percentages of gold, and need to cut each nugget into two parts so that the obtained parts can be alloyed into two ingots of weights $ 1$ kg ande $ 5$ kg containing the same proportion of gold. How we can do that?

2014 NZMOC Camp Selection Problems, 6

Tags: number theory , least common multiple , lcm

Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.

2020 Iranian Geometry Olympiad, 2

Tags: circumcircle , geometry , angle , rectangle , triangle

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

1994 AMC 8, 6

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The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is $\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

PEN O Problems, 33

Tags: induction

Assume that the set of all positive integers is decomposed into $r$ disjoint subsets $A_{1}, A_{2}, \cdots, A_{r}$ $A_{1} \cup A_{2} \cup \cdots \cup A_{r}= \mathbb{N}$. Prove that one of them, say $A_{i}$, has the following property: There exist a positive integer $m$ such that for any $k$ one can find numbers $a_{1}, \cdots, a_{k}$ in $A_{i}$ with $0 < a_{j+1}-a_{j} \le m \; (1\le j \le k-1)$.

2007 Romania National Olympiad, 3

Tags: combinatorics

The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$, with all vertices of the same colour.

2012 IMC, 3

Tags: group theory , abstract algebra

Given an integer $n>1$, let $S_n$ be the group of permutations of the numbers $1,\;2,\;3,\;\ldots,\;n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy? [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

2007 JBMO Shortlist, 1

Tags: function , quadratic , algebra , inequalities

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2019 Singapore Senior Math Olympiad, 1

Tags: circumcircle , angle bisector , geometry , parallelogram

In a parallelogram $ABCD$, the bisector of $\angle A$ intersects $BC$ at $M$ and the extension of $DC$ at $N$. Let $O$ be the circumcircle of the triangle $MCN$. Prove that $\angle OBC = \angle ODC$

1997 Vietnam Team Selection Test, 2

Tags: combinatorics , inequalities

There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied: 1) from each town there are exactly $ k$ direct routes to $ k$ other towns; 2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

2012 Finnish National High School Mathematics Competition, 5

Tags: function , number theory , induction

The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying \[ f(x)=\begin{cases} 3x+1,& \mbox{as }x\mbox{ is odd}\\ x/2, & \mbox{as }x\mbox{ is even.}\\ \end{cases} \] In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$ Prove that there is an $x\in\mathbb{Z}_+$ satisfying \[f^{40}(x)> 2012x.\]

2017 F = ma, 6

Tags: torque

6) In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 g$, $M_2 = 200 g$, and $M_4 = 500 g$. What is the value of $M_3$ for the system to be in static equilibrium? A) 300 g B) 400 g C) 500 g D) 600 g E) 700 g

2002 Irish Math Olympiad, 4

Tags: floor function , algebra

Let $ \alpha\equal{}2\plus{}\sqrt{3}$. Prove that $ \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n}$ for all $ n \in \mathbb{N}_0$.

2023 AMC 12/AHSME, 16

Tags: number theory , chicken mcnugget theorem

In Coinland, there are three types of coins, each worth $6,$ $10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have? $\textbf{(A) }11\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$ (I forgot the order)

2022 EGMO, 4

Tags: algebra

Given a positive integer $n \ge 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_0, a_1, \dots, a_N$ such that $(1) \ $ $a_0+a_1 = -\frac{1}{n},$ and $(2) \ $ $(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}$ for $1 \le k \le N-1$.

2002 AMC 8, 19

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How many whole numbers between 99 and 999 contain exactly one 0? $\text{(A)}\ 72 \qquad \text{(B)}\ 90 \qquad \text{(C)}\ 144 \qquad \text{(D)}\ 162 \qquad \text{(E)}\ 180$

2018 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry , distance , square

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

2022 Kyiv City MO Round 2, Problem 3

Tags: algebra

Nonzero real numbers $x_1, x_2, \ldots, x_n$ satisfy the following condition: $$x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}$$ Determine all $n$ for which $x_1, x_2, \ldots, x_n$ have to be equal. [i](Proposed by Oleksii Masalitin, Anton Trygub)[/i]

2007 AMC 12/AHSME, 13

Tags: probability

A traffic light runs repeatedly through the following cycle: green for $ 30$ seconds, then yellow for $ 3$ seconds, and then red for $ 30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? $ \textbf{(A)}\ \frac {1}{63}\qquad \textbf{(B)}\ \frac {1}{21}\qquad \textbf{(C)}\ \frac {1}{10}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {1}{3}$

1941 Moscow Mathematical Olympiad, 071

Tags: geometry , triangle construction

Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.

2001 India Regional Mathematical Olympiad, 7

Tags: number theory , modular arithmetic

Prove that the product of the first $1000$ positive even integers differs from the product of the first $1000$ positive odd integers by a multiple of $2001$.

2008 Purple Comet Problems, 3

Tags: percent

There were 891 people voting at precinct 91. There were 20 percent more female voters than male voters. How many female voters were there?

2023 ELMO Shortlist, C5

Tags: combinatorics

Define the [i]mexth[/i] of $k$ sets as the $k$th smallest positive integer that none of them contain, if it exists. Does there exist a family $\mathcal F$ of sets of positive integers such that [list] [*]for any nonempty finite subset $\mathcal G$ of $\mathcal F$, the mexth of $\mathcal G$ exists, and [*]for any positive integer $n$, there is exactly one nonempty finite subset $\mathcal G$ of $\mathcal F$ such that $n$ is the mexth of $\mathcal G$. [/list] [i]Proposed by Espen Slettnes[/i]