This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 119

2022 SAFEST Olympiad, 1
Tags: summation , number theory , algebra , floor function
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?

1999 Croatia National Olympiad, Problem 3
Tags: sum , summation , fibonacci , fibonacci sequence , algebra
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.

2019 Jozsef Wildt International Math Competition, W. 29
Tags: summation , integration
Prove that $$\int \limits_0^{\infty} e^{3t}\frac{4e^{4t}(3t - 1) + 2e^{2t}(15t - 17) + 18(1 - t)}{\left(1 + e^{4t} - e^{2t}\right)^2}=12\sum \limits_{k=0}^{\infty}\frac{(-1)^k}{(2k + 1)^2}-10$$

2011 VJIMC, Problem 3
Tags: summation
Prove that $$\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}$$for all $x\in(-1,1)$.

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P2
Tags: inequality , n variables , strict , summation , algebra
Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that $\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$. [i]Proposed by Mirko Petrusevski[/i]

2002 IMC, 9
Tags: summation , real analysis
For each $n\geq 1$ let $$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$ Show that $a_{n}\cdot b_{n}$ is an integer.

1991 IMO Shortlist, 11
Tags: series summation , summation , combinatorial identity , binomial coefficient , combinatorics
Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

1968 Yugoslav Team Selection Test, Problem 5
Tags: algebra , summation , polynomial
Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$. (a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$. (b) Do there exist integers $x,n$ for which $S(x,n)=0$?

2007 Germany Team Selection Test, 1
Tags: sequence , summation , floor function , number theory , calculus
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

1968 IMO, 6
Tags: summation , equation , floor function , number theory
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]

2019 Jozsef Wildt International Math Competition, W. 7
Tags: integration , summation , harmonic numbers , limit , sequence
If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$

1990 IMO Longlists, 2
Tags: series summation , summation , combinatorial identity , binomial coefficient , combinatorics
Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

2013 VJIMC, Problem 4
Tags: summation , binomial , algebra
Let $n$ and $k$ be positive integers. Evaluate the following sum $$\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}$$where $\binom nk=\frac{n!}{k!(n-k)!}$.

2021 AMC 10 Fall, 22
Tags: summation , algebra
For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $? $\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$

1983 IMO Longlists, 60
Tags: summation , series summation , approximation , inequality , algebra
Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

1980 Austrian-Polish Competition, 4
Tags: counting , combinatorics , summation , induction
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

2003 Kurschak Competition, 3
Tags: phi function , analytic number theory , expected value , probabilistic method , summation , number theory , inequalities
Prove that the following inequality holds with the exception of finitely many positive integers $n$: \[\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.\]

1975 IMO Shortlist, 7
Tags: algebra , combinatorics , summation , binomial coefficient identity , binomial coefficient
Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

1984 Putnam, A2
Tags: real analysis , summation , algebra
Express $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number.

2021 Indonesia TST, C
Tags: combinatorics , subset , counting , summation , prime
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 $p$.

2019 LIMIT Category A, Problem 5
Tags: trigonometry , summation , algebra
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$. $\textbf{(A)}~\frac n2$ $\textbf{(B)}~n$ $\textbf{(C)}~n\pi$ $\textbf{(D)}~\frac{n\pi}2$

2019 LIMIT Category B, Problem 10
Tags: summation , algebra
$\frac1{1+\sqrt3}+\frac1{\sqrt3+\sqrt5}+\frac1{\sqrt5+\sqrt7}+\ldots+\frac1{\sqrt{2017}+\sqrt{2019}}=?$ $\textbf{(A)}~\frac{\sqrt{2019}-1}2$ $\textbf{(B)}~\frac{\sqrt{2019}+1}2$ $\textbf{(C)}~\frac{\sqrt{2019}-1}4$ $\textbf{(D)}~\text{None of the above}$

2021 IMO Shortlist, A2
Tags: number theory , summation , algebra , floor function
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?

2024 AMC 12/AHSME, 21
Tags: summation , recursion
Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\] for all $n \ge 2.$ What is the greatest integer less than or equal to \[\sum^{100}_{n=1} a_n^2?\] $\textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \qquad \textbf{(D) } 338{,}553 \qquad \textbf{(E) } 338{,}554$

1989 French Mathematical Olympiad, Problem 5
Tags: summation , algebra
Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Denote $$s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}.$$ (a) Let $\lambda>1$ be a real number. Show that $s'<\lambda s+\frac\lambda{\lambda-1}$. (b) Deduce that $\sqrt{s'}<\sqrt s+1$.