Found problems: 15460
2016 Latvia National Olympiad, 3
Prove that among any 18 consecutive positive 3-digit numbers, there is at least one that is divisible by the sum of its digits!
2014 AIME Problems, 11
In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.
2023 Balkan MO Shortlist, N1
For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$ can be written as sum of two squares.
2005 IMO, 2
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.
Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
2010 Iran MO (3rd Round), 6
$g$ and $n$ are natural numbers such that $gcd(g^2-g,n)=1$ and $A=\{g^i|i \in \mathbb N\}$ and $B=\{x\equiv (n)|x\in A\}$(by $x\equiv (n)$ we mean a number from the set $\{0,1,...,n-1\}$ which is congruent with $x$ modulo $n$). if for $0\le i\le g-1$
$a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$
prove that $g-1|\sum_{i=0}^{g-1}ia_i$.( the symbol $|$ $|$ means the number of elements of the set)($\frac{100}{6}$ points)
the exam time was 4 hours
2025 Abelkonkurransen Finale, 2a
A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.
2016 China Team Selection Test, 4
Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that
$$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$
holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).
2019 Middle European Mathematical Olympiad, 4
Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$.
[i]Proposed by Kartal Nagy, Hungary[/i]
2024 International Zhautykov Olympiad, 3
Positive integer $d$ is not perfect square. For each positive integer $n$, let $s(n)$ denote the number of digits $1$ among the first $n$ digits in the binary representation of $\sqrt{d}$ (including the digits before the point). Prove that there exists an integer $A$ such that $s(n)>\sqrt{2n}-2$ for all integers $n\ge A$
2007 Thailand Mathematical Olympiad, 11
Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.
2023 Belarus Team Selection Test, 1.3
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
2008 AIME Problems, 13
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
2013 IFYM, Sozopol, 5
Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.
2016 Lusophon Mathematical Olympiad, 5
A numerical sequence is called lusophone if it satisfies the following three conditions:
i) The first term of the sequence is number $1$.
ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$.
(iii) The last term of the sequence is the number $2016$.
For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$
How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?
2009 Postal Coaching, 2
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$
2021 JBMO TST - Turkey, 5
$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.
2004 Olympic Revenge, 5
$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n.
prove that $a_n$ is one integer for all positive integers n.
1995 National High School Mathematics League, 2
Find all real number $p$, such that the three roots of the equation $5x^3-5(p+1)x^2+(71p-1)x+1=66p$ are all positive integers.
2009 Polish MO Finals, 2
Let $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements $ A,B$ of the subset there exists a point $ C$ contained in $ S$ such that the area of triangle $ ABC$ is equal to k .
2019 Durer Math Competition Finals, 5
We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?
1987 China Team Selection Test, 2
Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.
1994 Romania TST for IMO, 2:
Let $ n$ be an odd positive integer. Prove that $((n-1)^n+1)^2$ divides $ n(n-1)^{(n-1)^n+1}+n$.
2020 CMIMC Algebra & Number Theory, 9
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x,y)$ such that $1\le x,y \le p$ and $x^3-3xy+y^3+1$ is divisible by $p$.
2021 AIME Problems, 14
For any positive integer $a,$ $\sigma(a)$ denotes the sum of the positive integer divisors of $a.$ Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a.$ Find the sum of the prime factors in the prime factorization of $n.$
2014 District Olympiad, 2
For each positive integer $n$ we denote by $p(n)$ the greatest square less than or equal to $n$.
[list=a]
[*]Find all pairs of positive integers $( m,n)$, with $m\leq n$, for which
\[ p( 2m+1) \cdot p( 2n+1) =400 \]
[*]Determine the set $\mathcal{P}=\{ n\in\mathbb{N}^{\ast}\vert n\leq100\text{ and }\dfrac{p(n+1)}{p(n)}\notin\mathbb{N}^{\ast}\}$[/list]