Olympiad problems

2019 Math Prize for Girls Problems, 6

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For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.

1999 AMC 12/AHSME, 8

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At the end of $ 1994$, Walter was half as old as his grandmother. The sum of the years in which they were born was $ 3838$. How old will Walter be at the end of $ 1999$? $ \textbf{(A)}\ 48 \qquad \textbf{(B)}\ 49\qquad \textbf{(C)}\ 53\qquad \textbf{(D)}\ 55\qquad \textbf{(E)}\ 101$

2019 Costa Rica - Final Round, 1

In a faraway place in the Universe, a villain has a medal with special powers and wants to hide it so that no one else can use it. For this, the villain hides it in a vertex of a regular polygon with $2019$ sides. Olcoman, the savior of the Olcomita people, wants to get the medal to restore peace in the Universe, for which you have to pay $1000$ olcolones for each time he makes the following move: on each turn he chooses a vertex of the polygon, which turns green if the medal is on it or in one of the four vertices closest to it, or otherwise red. Find the fewest olcolones Olcoman needs to determine with certainty the position of the medal.

2016 Kurschak Competition, 2

Tags: combinatorics

Prove that for any finite set $A$ of positive integers, there exists a subset $B$ of $A$ satisfying the following conditions: [list][*]if $b_1,b_2\in B$ are distinct, then neither $b_1$ and $b_2$ nor $b_1+1$ and $b_2+1$ are multiples of each other, and [*] for any $a\in A$, we can find a $b\in B$ such that $a$ divides $b$ or $b+1$ divides $a+1$.[/list]

2021 Princeton University Math Competition, B1

Tags: combinatorics

A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.

2017 Greece Team Selection Test, 1

Tags: geometry

Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$, and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at $F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals $AEFA', BDFB', CDEC'$ are inscribable. (1) Prove that $DEA'B'$ is inscribable. (2) Prove that $DA', EB', FC'$ are concurrent.

2017 South East Mathematical Olympiad, 6

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $O$, where $AC\perp BD$. $M$ be the midpoint of arc $ADC$. Circle $(DOM)$ intersect $DA,DC$ at $E,F$. Prove that $BE=BF$.

2007 Indonesia TST, 4

Tags: combinatorics

Let $ n$ and $ k$ be positive integers. Please, find an explicit formula for \[ \sum y_1y_2 \dots y_k,\] where the summation runs through all $ k\minus{}$tuples positive integers $ (y_1,y_2,\dots,y_k)$ satisfying $ y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n$.

2008 Kurschak Competition, 2

Tags: combinatorics

Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$, and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$. (Here, the powers of $2$ are $1,2,4,\dots$.) Prove that $|S|\le |T|$.

1962 All Russian Mathematical Olympiad, 013

Tags: geometry , area

Given points $A' ,B' ,C' ,D',$ on the extension of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $$[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.

1997 Pre-Preparation Course Examination, 1

Tags: function , algebra

Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and \[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\] Show that $f$ is a periodic function.

2006 Germany Team Selection Test, 1

Tags: combinatorics

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

1995 Baltic Way, 19

Tags: geometry

The following construction is used for training astronauts: A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?

2022 Durer Math Competition Finals, 16

Tags: number theory , divisor

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

2019 CCA Math Bonanza, L5.3

Tags: function

For a positive integer $n$, let $d\left(n\right)$ be the number of positive divisors of $n$ (for example $d\left(39\right)=4$). Estimate the average value that $d\left(n\right)$ takes on as $n$ ranges from $1$ to $2019$. An estimate of $E$ earns $2^{1-\left|A-E\right|}$ points, where $A$ is the actual answer. [i]2019 CCA Math Bonanza Lightning Round #5.3[/i]

2015 NIMO Problems, 3

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How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.) [i]Based on a proposal by Yannick Yao[/i]

1990 IMO Longlists, 52

Tags: inequalities

Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \ i = 1, 2, \ldots, n$. Prove that (i) If $n = 4$, then \[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.\] (ii) If $n = 6$, then \[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.\]

2023 Junior Balkan Team Selection Tests - Moldova, 6

Tags: algebra

Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$.

1997 IMC, 1

Tags: limit , calculus , integration , logarithm , real analysis

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

2006 Taiwan National Olympiad, 1

Tags: circumcircle , angle bisector , geometry

$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.

2009 Jozsef Wildt International Math Competition, W. 5

Tags: number theory , prime number

Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.

2021 Sharygin Geometry Olympiad, 8.2

Tags: geometry , reflection , concurrency

Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.

2010 Contests, 2

Tags: number theory , remainder

Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.

2001 AIME Problems, 8

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Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double?

2008 Bulgaria Team Selection Test, 3

Tags: function , algebra

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.