Found problems: 119
2019 Jozsef Wildt International Math Competition, W. 54
Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$
2019 LIMIT Category C, Problem 7
The value of
$$\left(1+\frac26+\frac{2\cdot5}{6\cdot12}+\frac{2\cdot5\cdot8}{6\cdot12\cdot18}+\ldots\right)^3$$
2013 BMT Spring, P1
Prove that for all positive integers $m$ and $n$,
$$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$
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:
\[
\left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024
\]
1989 IMO Longlists, 35
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$
1974 IMO, 3
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
1978 IMO, 2
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
2019 LIMIT Category A, Problem 5
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$.
$\textbf{(A)}~\frac n2$
$\textbf{(B)}~n$
$\textbf{(C)}~n\pi$
$\textbf{(D)}~\frac{n\pi}2$
2022 Saudi Arabia IMO TST, 1
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?
2021 Indonesia TST, C
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
1978 Putnam, B2
Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$
as a rational number.
2013 Hanoi Open Mathematics Competitions, 5
The number $n$ is called a composite number if it can be written in the form $n = a\times b$, where $a, b$ are positive
integers greater than $1$. Write number $2013$ in a sum of $m$ composite numbers. What is the largest value of $m$?
(A): $500$, (B): $501$, (C): $502$, (D): $503$, (E): None of the above.
1997 VJIMC, Problem 4-M
Prove that
$$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$
2015 VJIMC, 3
[b]Problem 3[/b]
Determine the set of real values of $x$ for which the following series converges, and find its sum:
$$\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ . $$
II Soros Olympiad 1995 - 96 (Russia), 11.7
Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.
2021 IMO Shortlist, A2
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?
2021 Harvard-MIT Mathematics Tournament., 4
Let $k$ and $n$ be positive integers and let
$$S=\{(a_1,\ldots,a_k)\in \mathbb{Z}^{k}\;|\; 0\leq a_k\leq\cdots\leq a_1 \leq n,a_1+\cdots+a_k=n\}$$.
Determine, with proof, the value of
$$\sum_{(a_1,\ldots,a_k)\in S}\binom{n}{a_1}\binom{a_1}{a_2}\cdots\binom{a_{k-1}}{a_k}$$
in terms of $k$ and $n$, where the sum is over all $k$-tuples in $S$.
1983 IMO Shortlist, 21
Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$
2020 Jozsef Wildt International Math Competition, W51
Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that
$$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$
Show that:
$$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$
[i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]
1995 IMC, 8
Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$,
$b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$. Calculate
$$\sum_{n=1}^{\infty}b_{n}2^{n}.$$
2010 VTRMC, Problem 7
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
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]
2019 LIMIT Category B, Problem 10
$\frac1{1+\sqrt3}+\frac1{\sqrt3+\sqrt5}+\frac1{\sqrt5+\sqrt7}+\ldots+\frac1{\sqrt{2017}+\sqrt{2019}}=?$
$\textbf{(A)}~\frac{\sqrt{2019}-1}2$
$\textbf{(B)}~\frac{\sqrt{2019}+1}2$
$\textbf{(C)}~\frac{\sqrt{2019}-1}4$
$\textbf{(D)}~\text{None of the above}$
1974 IMO Longlists, 30
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
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]