Found problems: 15460
2014 PUMaC Number Theory B, 8
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
2021 Serbia JBMO TSTs, 2
Solve the following equation in natural numbers:
\begin{align*}
x^2=2^y+2021^z
\end{align*}
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2021 Belarusian National Olympiad, 8.6
For four pairwise different positive integers $a,b,c$ and $d$ six numbers are calculated: $ab+10$,$ac+10$,$ad+10$,$bc+10$,$bd+10$ and $cd+10$.
Find the maximum amount of them which can be perfect squares.
2016 CHMMC (Fall), 3
For a positive integer $m$, let $f(m)$ be the number of positive integers $q \le m$ such that $\frac{q^2-4}{m}$ is an integer. How many positive square-free integers $m < 2016$ satisfy $f(m) \ge 16$?
2004 Estonia National Olympiad, 4
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.
2015 IFYM, Sozopol, 7
Determine the greatest natural number $n$, such that for each set $S$ of 2015 different integers there exist 2 subsets of $S$ (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by $n$.
2022 Bulgaria JBMO TST, 1
Are there positive integers $a$, $b$, $c$ and $d$ such that:
a) $a^{2021} + b^{2023} = 11(c^{2022} + d^{2024})$;
b) $a^{2022} + b^{2022} = 11(c^{2022} + d^{2022})$?
2000 Italy TST, 1
Determine all triples $(x,y,z)$ of positive integers such that
\[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]
2022 CMWMC, R5
[u]Set 5[/u]
[b]p13.[/b] An equiangular $12$-gon has side lengths that alternate between $2$ and $\sqrt3$. Find the area of the circumscribed circle of this $12$-gon.
[b]p14.[/b] For positive integers $n$, let $\sigma(n)$ denote the number of positive integer factors of $n$. Then $\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64$. Let $S$ be the set of factors $k$ of $17280$ such that $\sigma(k) = 32$. If $p$ is the product of the elements of $S$, find $\sigma(p)$.
[b]p15.[/b] How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid.
PS. You should use hide for answers.
2014 Poland - Second Round, 6.
Call a positive number $n$ [i]fine[/i], if there exists a prime number $p$ such that $p|n$ and $p^2\nmid n$. Prove that at least 99% of numbers $1, 2, 3, \ldots, 10^{12}$ are fine numbers.
2008 Purple Comet Problems, 14
Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that $\frac{M}{N}$ is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.
2021 BMT, 2
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?
2018 Thailand TST, 1
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
1985 Tournament Of Towns, (084) T5
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ?
(S. Fomin , Leningrad)
1990 Swedish Mathematical Competition, 6
Find all positive integers $m, n$ such that $\frac{117}{158} > \frac{m}{n} > \frac{97}{131}$ and $n \le 500$.
2017 Princeton University Math Competition, A1/B3
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compute $M - m$.
1995 Tournament Of Towns, (468) 2
The first five terms of a sequence are $1, 2, 3, 4$ and $5$. From the sixth term on, each term is $1$ less than the product of all the proceeding ones. Prove that the product of the first$ 70$ terms is equal to the sum of their squares.
(LD Kurliandchik)
1995 Italy TST, 1
Determine all triples $(x,y,z)$ of integers greater than $1$ with the property that $x$ divides $yz-1$, $y$ divides $zx-1$ and $z$ divides $xy-1$.
2016 Japan Mathematical Olympiad Preliminary, 1
Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.
2016 Hong Kong TST, 1
Find all prime numbers $p$ and $q$ such that $p^2|q^3+1$ and $q^2|p^6-1$
2004 Federal Math Competition of S&M, 2
The sequence $(a_n)$ is determined by $a_1 = 0$ and
$(n+1)^3a_{n+1} = 2n^2(2n+1)a_n+2(3n+1)$ for $n \geq 1$.
Prove that infinitely many terms of the sequence are positive integers.
PEN A Problems, 19
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.
2010 Mathcenter Contest, 2
A positive rational number $x$ is called [i]banzai [/i] if the following conditions are met:
$\bullet$ $x=\frac{p}{q}>1$ where $p,q$ are comprime natural numbers
$\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$
Find the total number of banzai numbers.
Note:$\left\{\,x\right\}$ means fractional part of $x$
[i](tatari/nightmare)[/i]
2012 BMT Spring, 8
Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ ... \circ \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = min \{k \in N | \phi^k (n)=1 \}$
. For example,
$\phi^1 (13) = \phi(13) = 12$
$\phi^2 (13) = \phi (\phi (13)) = 4$
$\phi^3 (13) = \phi(\phi(\phi(13))) = 2$
$\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$
so $\theta (13) = 4$. Let $f(r) = \theta (13^r)$. Determine $f(2012)$.