Found problems: 119
1978 IMO Shortlist, 6
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
2010 ELMO Shortlist, 1
For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$.
[list=1]
[*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?
[*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list]
[i]Brian Hamrick.[/i]
1966 IMO Longlists, 29
A given natural number $N$ is being decomposed in a sum of some consecutive integers.
[b]a.)[/b] Find all such decompositions for $N=500.$
[b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only?
[b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$
[b]Note by Darij:[/b] The $0$ is not considered as a natural number.
2011 VJIMC, Problem 2
Let $k$ be a positive integer. Compute
$$\sum_{n_1=1}^\infty\sum_{n_2=1}^\infty\cdots\sum_{n_k=1}^\infty\frac1{n_1n_2\cdots n_k(n_1+n_2+\ldots+n_k+1)}.$$
1968 IMO Shortlist, 15
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
1967 IMO Shortlist, 3
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
2025 Romania EGMO TST, P1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
1982 Putnam, A2
For positive real $x$, let
$$B_n(x)=1^x+2^x+\ldots+n^x.$$Prove or disprove the convergence of
$$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$
2018 Brazil Undergrad MO, 24
What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$
2019 India PRMO, 13
Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i <
j \leq 101} x_i x_j$ ?
1981 Putnam, B1
Find
$$\lim_{n\to \infty} \frac{1}{n^5 } \sum_{h=1}^{n} \sum_{k=1}^{n} (5h^4 -18h^2 k^2 +5k^4).$$
2019 Jozsef Wildt International Math Competition, W. 55
Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$
2022 Brazil Team Selection Test, 2
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2019 Jozsef Wildt International Math Competition, W. 61
If $a$, $b$, $c \in \mathbb{R}$ then$$\sum \limits_{cyc} \sqrt{(c+a)^2b^2+c^2a^2}+\sqrt{5}\left |\sum \limits_{cyc} \sqrt{ab}\right |\geq \sum \limits_{cyc}\sqrt{(ab+2bc+ca)^2+(b+c)^2a^2}$$
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
1974 IMO Shortlist, 6
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
2020 Jozsef Wildt International Math Competition, W41
If $m,n\in\mathbb N_{\ge2}$, find the best constant $k\in\mathbb R$ for which
$$\sum_{j=2}^m\sum_{i=2}^n\frac1{i^j}<k$$
[i]Proposed by Dorin Mărghidanu[/i]
1981 Putnam, B5
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether
$$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$
is a rational number.
1996 VJIMC, Problem 2
Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether
$$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$
2019 Jozsef Wildt International Math Competition, W. 25
Let $x_i$, $y_i$, $z_i$, $w_i \in \mathbb{R}^+, i = 1, 2,\cdots n$, such that$$\sum \limits_{i=1}^nx_i=nx,\ \sum \limits_{i=1}^ny_i=ny,\ \sum \limits_{i=1}^nw_i=nw $$ $$\Gamma \left(z_i\right)\geq \Gamma \left(w_i\right),\ \sum \limits_{i=1}^n\Gamma \left(z_i\right)=n\Gamma^* (z)$$Then$$\sum \limits_{i=1}^n \frac{\left(\Gamma \left(x_i\right)+\Gamma \left(y_i\right)\right)^2}{\Gamma \left(z_i\right)-\Gamma \left(w_i\right)}\geq n\frac{\left(\Gamma \left(x\right)+\Gamma \left(y\right)\right)^2}{\Gamma^* \left(z\right)-\Gamma \left(w\right)}$$
1989 IMO Longlists, 35
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$
2013 BMT Spring, P1
Prove that for all positive integers $m$ and $n$,
$$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$
2015 Hanoi Open Mathematics Competitions, 15
Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$.
Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$
2024 CIIM, 3
Given a positive integer \(n\), let \(\phi(n)\) denote the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). Find all possible positive integers \(k\) for which there exist positive integers \(1 \leq a_1 < a_2 < \dots < a_k\) such that:
\[
\left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024
\]