Found problems: 15460
2009 Portugal MO, 1
A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?
2024 Middle European Mathematical Olympiad, 7
Define [i]glueing[/i] of positive integers as writing their base ten representations one after another and
interpreting the result as the base ten representation of a single positive integer.
Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \ge N_k$, we can glue the numbers $1,2,\dots,n$ in some order so that the result is a number divisible by $k$.
[i]Remark[/i]. The base ten representation of a positive integer never starts with zero.
[i]Example[/i]. Glueing $15, 14, 7$ in this order makes $15147$.
2017 Bulgaria EGMO TST, 1
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions
\[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\]
for all $x \in \mathbb{Q^+}.$
2014 Thailand TSTST, 3
Let $s(n)$ denote the sum of digits of a positive integer $n$. Prove that $s(9^n) > 9$ for all $n\geq 3$.
2006 ISI B.Math Entrance Exam, 8
Let $S$ be the set of all integers $k$, $1\leq k\leq n$, such that $\gcd(k,n)=1$. What is the arithmetic mean of the integers in $S$?
2016 Korea Junior Math Olympiad, 4
find all positive integer $n$, satisfying
$$\frac{n(n+2016)(n+2\cdot 2016)(n+3\cdot 2016) . . . (n+2015\cdot 2016)}{1\cdot 2 \cdot 3 \cdot . . . . . \cdot 2016}$$ is positive integer.
2003 Croatia Team Selection Test, 1
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2010 Saudi Arabia IMO TST, 2
a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$
b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient
2011 Bosnia And Herzegovina - Regional Olympiad, 2
For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$
2018 Costa Rica - Final Round, N2
Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$
2016 JBMO Shortlist, 5
Determine all four-digit numbers $\overline{abcd} $ such that
$(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd} $:
2017 Princeton University Math Competition, A5/B7
Let $p(n) = n^4-6n^2-160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)
Brazil L2 Finals (OBM) - geometry, 2015.3
Let $ABC$ be a triangle and $n$ a positive integer. Consider on the side $BC$ the points $A_1, A_2, ..., A_{2^n-1}$ that divide the side into $2^n$ equal parts, that is, $BA_1=A_1A_2=...=A_{2^n-2}A_{2^n-1}=A_{2^n-1}C$. Set the points $B_1, B_2, ..., B_{2^n-1}$ and $C_1, C_2, ..., C_{2^n-1}$ on the sides $CA$ and $AB$, respectively, analogously. Draw the line segments $AA_1, AA_2, ..., AA_{2^n-1}$, $BB_1, BB_2, ..., BB_{2^n-1}$ and $CC_1, CC_2, ..., CC_{2^n-1}$. Find, in terms of $n$, the number of regions into which the triangle is divided.
1987 IMO Longlists, 54
Let $n$ be a natural number. Solve in integers the equation
\[x^n + y^n = (x - y)^{n+1}.\]
2015 Belarus Team Selection Test, 1
Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$.
B.Gilevich
2023 IRN-SGP-TWN Friendly Math Competition, 1
Prove that there are infinitely many positive integers $a$ such that \[a!+(a+2)!\mid (a+2\left\lfloor\sqrt{a}\right\rfloor)!.\]
[i]Proposed by Navid and the4seasons.[/i]
1985 Greece National Olympiad, 3
Consider the line (E): $5x-10y+3=0$ . Prove that:
a) Line $(E)$ doesn't pass through points with integer coordinates.
b) There is no point $A(a_1,a_2)$ with $ a_1,a_2 \in \mathbb{Z}$ with distance from $(E)$ less then $\frac{\sqrt3}{20}$.
1992 Rioplatense Mathematical Olympiad, Level 3, 6
Definition: A natural number is [i]abundant [/i] if the sum of its positive divisors is greater than its double.
Find an odd abundant number and prove that there are infinitely many odd abundant numbers.
2000 All-Russian Olympiad Regional Round, 11.3
Sequence of real numbers $a_1, a_2, . . . , a_{2000}$ is such that for any natural number $n$, $1\le n \le 2000$, the equality $$a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.$$ Prove that all terms of this sequence are integers.
2016 Middle European Mathematical Olympiad, 4
Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$.
Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.
2008 VJIMC, Problem 3
Find all pairs of natural numbers $(n,m)$ with $1<n<m$ such that the numbers $1$, $\sqrt[n]n$ and $\sqrt[m]m$ are linearly dependent over the field of rational numbers $\mathbb Q$.
2011 HMNT, 8
Find the number of integers $x$ such that the following three conditions all hold:
$\bullet$ $x$ is a multiple of $5$
$\bullet$ $121 < x < 1331$
$\bullet$ When $x$ is written as an integer in base $11$ with no leading $0$s (i.e. no $0$s at the very left), its rightmost digit is strictly greater than its leftmost digit.
1988 Brazil National Olympiad, 1
Find all primes which are sum of two primes and difference of two primes.
2006 Thailand Mathematical Olympiad, 5
Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$
2013 AMC 12/AHSME, 23
$ ABCD$ is a square of side length $ \sqrt{3} + 1 $. Point $ P $ is on $ \overline{AC} $ such that $ AP = \sqrt{2} $. The square region bounded by $ ABCD $ is rotated $ 90^{\circ} $ counterclockwise with center $ P $, sweeping out a region whose area is $ \frac{1}{c} (a \pi + b) $, where $a $, $b$, and $ c $ are positive integers and $ \text{gcd}(a,b,c) = 1 $. What is $ a + b + c $?
$\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23 $