Olympiad problems

2004 Germany Team Selection Test, 3

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

2012 Indonesia TST, 1

Tags: induction , algebra , binary representation , sequence , recurrence relation

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

Russian TST 2019, P2

Tags: permutation , combinatorics

For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.

2018 Iran MO (1st Round), 18

Tags: geometry

Three rods of lengths $1396, 1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?

2021 Kyiv Mathematical Festival, 1

Tags: number theory

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

2000 China National Olympiad, 1

Tags: permutation , combinatorics , permutation statistics

Given an ordered $n$-tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$, we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the “[i]innovated tuple[/i]” of $A$. The number of distinct elements in $B$ is called the “[i]innovated degree[/i]” of $A$. Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$

2019 ASDAN Math Tournament, 2

Tags:

Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$?

2015 USAMTS Problems, 3

Tags:

Let $P$ be a convex n-gon in the plane with vertices labeled $V_1,...,V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles the blue and green regions are the same. Suppose $P$ has exactly one balancing point/ Show that the balancing point must be a vertex of $P$

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , root , trinomial

Let $a$ and $b$ be arbitrary distinct numbers. Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.

2024 Brazil Cono Sur TST, 3

Tags: combinatorics , set

For a pair of integers $a$ and $b$, with $0<a<b<1000$, a set $S\subset \begin{Bmatrix}1,2,3,...,2024\end{Bmatrix}$ $escapes$ the pair $(a,b)$ if for any elements $s_1,s_2\in S$ we have $\left|s_1-s_2\right| \notin \begin{Bmatrix}a,b\end{Bmatrix}$. Let $f(a,b)$ be the greatest possible number of elements of a set that escapes the pair $(a,b)$. Find the maximum and minimum values of $f$.

2023 Bulgaria EGMO TST, 2

Tags: number theory , greatest common divisor , function

Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.

2010 Miklós Schweitzer, 10

Tags: topology

Consider the space $ \{0,1 \} ^{N} $ with the product topology (where $\{0,1 \}$ is a discrete space). Let $ T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}} $ be the left-shift, ie $ (Tx) (n) = x (n+1) $ for every $ n \in \mathbb {N} $. Can a finite number of Borel sets be given: $ B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N} $ such that $$ \left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \} $$the $ \sigma $-algebra generated by the set system coincides with the Borel set system?

2005 France Pre-TST, 4

Tags: inequalities

Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2 = 25.$ Find the minimum of $\frac {xy} z + \frac {yz} x + \frac {zx} y .$ Pierre.

2021 AMC 12/AHSME Fall, 7

Tags: important announcement

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$ $\textbf{(B)}\: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

1955 Moscow Mathematical Olympiad, 314

Tags: polynomial , root , algebra

Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0$, cannot have two positive roots.

PEN H Problems, 43

Tags: diophantine equation

Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.

2014 India IMO Training Camp, 1

Tags: polynomial , number theory

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

2014 National Olympiad First Round, 11

Tags: quadratic

What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ -4 \qquad\textbf{(D)}\ -6 \qquad\textbf{(E)}\ -8 $

2008 Stanford Mathematics Tournament, 2

Tags:

How many primes exist which are less than 50?

2002 CentroAmerican, 1

Tags:

For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?

1999 Tournament Of Towns, 2

Tags: number theory , prime , sum of powers , sum

Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$. (a) May it happen that $d = 2$? (b) May it happen that $d$ is prime? (V Senderov)

2002 Junior Balkan Team Selection Tests - Moldova, 9

Tags: inequalities , algebra

The real numbers $a$ and $b$ satisfy the relation $a + b \ge 1$. Show that $8 (a^4 + b^4) \ge 1$.

2014 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry

$D$ is inner point of triangle $ABC$. $E$ is on $BD$ and $CE=BD$. $\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60$ Prove, that $AB>AC$

2019 South East Mathematical Olympiad, 6

Tags: algebra

Let $a,b,c$ be the lengths of the sides of a given triangle.If positive reals $x,y,z$ satisfy $x+y+z=1,$ find the maximum of $axy+byz+czx.$

1999 Cono Sur Olympiad, 4

Tags: number theory , coloring , multiple , digit

Let $A$ be a six-digit number, three of which are colored and equal to $1, 2$, and $4$. Prove that it is always possible to obtain a number that is a multiple of $7$, by performing only one of the following operations: either delete the three colored figures, or write all the numbers of $A$ in some order.