Olympiad problems

2003 AIME Problems, 11

Tags: trig identities , relatively prime , circumcircle , number theory , law of cosines , trigonometry , geometry

Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$

1998 IMO, 6

Tags: number theory , functional equation , prime number , algebra

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

1972 IMO Longlists, 32

Tags: number theory , inequalities

If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$, show that \[max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},\] where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$; i.e., $n = tk + r, 0 \le r \le k- 1$.

1993 China Team Selection Test, 1

Tags: modular arithmetic , number theory , greatest common divisor

Find all integer solutions to $2 x^4 + 1 = y^2.$

1969 IMO Longlists, 19

Tags: relatively prime , number theory , periodic sequence , divisibility

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

2023 ISL, N4

Tags: number theory , arithmetic sequence

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

1999 CentroAmerican, 2

Tags: number theory , modular arithmetic

Find a positive integer $n$ with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of $n$ into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number $m$ that is a divisor of $n$.

1999 Bundeswettbewerb Mathematik, 2

Tags: inequalities , number theory , sum of digits , digit

For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$. Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.

2009 Middle European Mathematical Olympiad, 11

Tags: number theory

Find all pairs $ (m$, $ n)$ of integers which satisfy the equation \[ (m \plus{} n)^4 \equal{} m^2n^2 \plus{} m^2 \plus{} n^2 \plus{} 6mn.\]

1959 Poland - Second Round, 4

Tags: perfect cube , number theory , divide

Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.

2019 Saudi Arabia IMO TST, 2

Tags: number theory , divisor

Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and a) Product of all divisor of $sm, sn$ are equal. b) Number of divisors of $sm,sn$ are equal.

2021 Baltic Way, 17

Tags: prime , number theory

Distinct positive integers $a, b, c, d$ satisfy $$\begin{cases} a \mid b^2 + c^2 + d^2,\\ b\mid a^2 + c^2 + d^2,\\ c \mid a^2 + b^2 + d^2,\\ d \mid a^2 + b^2 + c^2,\end{cases}$$ and none of them is larger than the product of the three others. What is the largest possible number of primes among them?

2021 Purple Comet Problems, 7

Tags: number theory

Find the sum of all positive integers $x$ such that there is a positive integer $y$ satisfying $9x^2 - 4y^2 = 2021$.

1996 Vietnam Team Selection Test, 2

Tags: floor function , number theory

For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i$. Find all $n$ such that $f(n) = 1996.$ [hide="old version"]For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}$. Find all $n$ such that $f(n) = 1996.$[/hide]

2010 China Second Round Olympiad, 2

Tags: number theory , modular arithmetic

Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer.

2010 All-Russian Olympiad, 1

Tags: number theory

If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.

2003 Italy TST, 1

Tags: number theory , modular arithmetic

Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

2005 Junior Balkan Team Selection Tests - Romania, 12

Tags: modular arithmetic , number theory

Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \] [i]Adrian Stoica[/i]

2018 Peru EGMO TST, 6

Tags: number theory

Find all positive integers $n$ such that the number $\frac{(2n)!+1}{n!+1}$ is positive integer.

2008 JBMO Shortlist, 6

Tags: number theory

Let $f : N \to R$ be a function, satisfying the following condition: for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \big(\frac{n}{p}\big)-f(p)$. If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$

2002 Germany Team Selection Test, 3

Tags: decimal representation , number theory , factorial , digit

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2017 Princeton University Math Competition, 12

Tags: number theory

Call a positive integer $n$ [i]tubular [/i] if for any two distinct primes $p$ and $q$ dividing $n, (p + q) | n$. Find the number of tubular numbers less than $100,000$. (Integer powers of primes, including $1, 3$, and $16$, are not considered [i]tubular[/i].)

2013 BMT Spring, 8

Tags: number theory

The three-digit prime number $p$ is written in base $2$ as $p_2$ and in base $5$ as $p_5$, and the two representations share the same last $2$ digits. If the ratio of the number of digits in $p_2$ to the number of digits in $p_5$ is $5$ to $2$, find all possible values of $p$.

2008 Baltic Way, 7

Tags: number theory , symmetry

How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?

2023 Indonesia TST, N

Tags: number theory

Find all triplets natural numbers $(a, b, c)$ satisfied \[GCD(a, b) + LCM(a,b) = 2021^c\] with $|a - b|$ and $(a+b)^2 + 4$ are both prime number