Found problems: 15460
2011 HMNT, 1
Find the number of positive integers $x$ less than $100$ for which $$3^x + 5^x + 7^x + 11^x + 13^x + 17^x + 19^x$$ is prime.
2024 Malaysian IMO Training Camp, 4
Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely?
[i](Proposed by Wong Jer Ren)[/i]
2006 USAMO, 5
A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$
2007 Pre-Preparation Course Examination, 22
Prove that for any positive integer $n \geq 3$ there exist positive integers $a_1,a_2,\cdots , a_n$ such that
\[a_1a_2\cdots a_n \equiv a_i \pmod {a_i^2} \qquad \forall i \in \{1,2,\cdots ,n\}\]
2013 AIME Problems, 6
Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as $\frac{m}{n}$, where $m$ and $n$ Are relatively prime positive integers. Find $m+n$.
1985 Poland - Second Round, 2
Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.
2022 Taiwan TST Round 3, N
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
2012 CHMMC Fall, 9
For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when $$\sum^{2^{12}-1}_{n=0}f(n)$$ is divided by $2^{12}$.
2001 AIME Problems, 6
A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2018 Nepal National Olympiad, 1a
[b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$.
[color=red]NOTE: There is a high chance that this problems was copied.[/color]
2006 Estonia Team Selection Test, 6
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.
Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form $(a, b)$ with $a\mid b$.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
2005 All-Russian Olympiad Regional Round, 8.5
It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.
2021 Regional Olympiad of Mexico West, 3
The sequence of real numbers $a_1, a_2, a_3, ...$ is defined as follows: $a_1 = 2019$, $a_2 = 2020$, $a_3 = 2021$ and for all $n \ge 1$
$$a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n.$$
Show that this sequence does not contain numbers of the form $m^6$ where $m$ is a positive integer.
2012 CHMMC Spring, 1
Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, ..., x_{10})$ of nonnegative integers such that
$$x^2_1+ x^2_2+ ... + x^2_{10} = k.$$
Let $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Find $\sum^{2012}_{i=1} b_{4i}$.
2023 Chile National Olympiad, 5
What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?
2013 NZMOC Camp Selection Problems, 8
Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.
1996 Austrian-Polish Competition, 1
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties:
(i) $n$ has exactly $k$ digits (in decimal representation),
(ii) all the digits of $n$ are odd,
(iii) $n$ is divisible by $5$,
(iv) the number $m = n/5$ has $k$ odd digits
2006 Switzerland - Final Round, 3
Calculate the sum of digit of the number
$$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$
where the number of nines doubles in each factor.
2006 MOP Homework, 4
Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.
1996 Nordic, 1
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.
2014 JBMO Shortlist, 1
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]
2022 BMT, 2
Compute the number of positive integer divisors of $100000$ which do not contain the digit $0.$
1986 Poland - Second Round, 4
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation
$$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$
Prove that $ x + y $ is the square of a natural number.
2005 China Team Selection Test, 1
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
2019 USA IMO Team Selection Test, 4
We say that a function $f: \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0} \to \mathbb{Z}$ is [i]great[/i] if for any nonnegative integers $m$ and $n$,
\[f(m + 1, n + 1) f(m, n) - f(m + 1, n) f(m, n + 1) = 1.\]
If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$).
Prove that if $A$, $B$, $C$, and $D$ are four sequences of integers satisfying $A \sim B$, $B \sim C$, and $C \sim D$, then $D \sim A$.
[i]Ankan Bhattacharya[/i]