Olympiad problems

2021 Argentina National Olympiad, 1

Tags: sequence , combinatorics

An infinite sequence of digits $1$ and $2$ is determined by the following two properties: i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained. In which position is the hundredth digit $1$? What is the thousandth digit of the sequence?

1974 IMO Shortlist, 3

Tags: coefficient , algebra , polynomial , number theory , degree

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

2010 Switzerland - Final Round, 1

Tags: combinatorics , induction

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $ 1$ unit to the right and the second one $ 1$ unit to the left. Under which initial conditions is it possible to move all coins to one single point?

2015 Saudi Arabia JBMO TST, 1

Tags: number theory , digit , divisible

A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$

2021 Girls in Math at Yale, 6

Tags: college

Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions: [list] [*] She rolled a $6$ exactly three times; [*] The product of her first three rolls is the same as the product of her last three rolls. [/list] How many distinct sequences of six rolls could Kara have rolled? [i]Proposed by Andrew Wu[/i]

1956 AMC 12/AHSME, 6

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In a group of cows and chickens, the number of legs was $ 14$ more than twice the number of heads. The number of cows was: $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 14$

1952 AMC 12/AHSME, 9

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If $ m \equal{} \frac {cab}{a \minus{} b}$, then $ b$ equals: $ \textbf{(A)}\ \frac {m(a \minus{} b)}{ca} \qquad\textbf{(B)}\ \frac {cab \minus{} ma}{ \minus{} m} \qquad\textbf{(C)}\ \frac {1}{1 \plus{} c} \qquad\textbf{(D)}\ \frac {ma}{m \plus{} ca}$ $ \textbf{(E)}\ \frac {m \plus{} ca}{ma}$

2019 AMC 8, 7

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Shauna takes $5$ tests, each worth a maximum of a $100$ points. Her scores on the first three tests were $76$, $94$, and $87$. In order to average an $81$ on all five tests, what is the lowest score she could earn on one of the two tests? $\textbf{(A) } 48 \qquad\textbf{(B) } 52 \qquad\textbf{(C) } 66 \qquad\textbf{(D) } 70 \qquad\textbf{(E) } 74$

2005 Germany Team Selection Test, 3

Tags: inequalities , geometry

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: combinatorics , combinatorial number theory , coprime , set

Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that: every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number

2021 Simon Marais Mathematical Competition, B3

Tags: calculus , integration , function

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties. (i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$. (ii) For every real number $x$ and every integer $n \ge 1$ we have \[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]

2005 Miklós Schweitzer, 5

Tags: linear algebra , archimedean , compactness

Let $GL(n, K)$ be a linear group over the field K with a topology induced by a non-Archimedean absolute value of the field K. Prove that if the matrix $M \in GL (n, K)$ is contained by some compact subgroup of $GL(n, K)$, then all eigenvalues of M have absolute value 1.

2012 Math Prize For Girls Problems, 18

Tags: probability , expected value

Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will take?

2018 Azerbaijan BMO TST, 1

Tags: inequalities , algebra

Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$

1986 AIME Problems, 8

Tags: logarithm

Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$?

2016 May Olympiad, 3

Tags: digit , number theory , divide

We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero. a) Find a quad-divi number such that the sum of its digits is $24$. b) Find a quad-divi number such that the sum of its digits is $1001$.

2010 Contests, 1

Tags: modular arithmetic , number theory

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

1979 Miklós Schweitzer, 7

Tags: geometry , 3d geometry , sphere , vector , combinatorics

Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist. [i]L. Lovasz[/i]

2020 Purple Comet Problems, 28

Tags: number theory

Let $p, q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.

2017 Online Math Open Problems, 17

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For a positive integer $n$, define $f(n)=\sum_{i=0}^{\infty}\frac{\gcd(i,n)}{2^i}$ and let $g:\mathbb N\rightarrow \mathbb Q$ be a function such that $\sum_{d\mid n}g(d)=f(n)$ for all positive integers $n$. Given that $g(12321)=\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $v_2(p)$. [i]Proposed by Michael Ren[/i]

2018 Iran MO (1st Round), 21

Tags: geometry , geometric transformation , reflection

The point $P$ is chosen inside or on the equilateral triangle $ABC$ of side length $1$. The reflection of $P$ with respect to $AB$ is $K$, the reflection of $K$ about $BC$ is $M$, and the reflection of $M$ with respect to $AC$ is $N$. What is the maximum length of $NP$? $\textbf{(A)}\ 2\sqrt 3\qquad\textbf{(B)}\ \sqrt 3\qquad\textbf{(C)}\ \frac{\sqrt 3}{2} \qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 1$

2006 Balkan MO, 1

Tags: rearrangement inequality , inequalities theorems , inequalities

Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality \[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]

2013 India PRMO, 1

Tags: number theory , minimum

What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

2006 Taiwan TST Round 1, 2

Tags: floor function , geometry , rectangle , polynomial , quadratic , number theory

Let $p,q$ be two distinct odd primes. Calculate $\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.

2021 AMC 12/AHSME Fall, 8

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The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s height to the base. What is the measure, in degrees, of the vertex angle of this triangle? $\textbf{(A)}\ 105 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 135 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 165$