Olympiad problems

1975 IMO Shortlist, 7

Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

2015 Hanoi Open Mathematics Competitions, 15

Tags: algebra , maximum , summation , inequalities

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$

1981 Putnam, B1

Tags: limit , summation

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. 8

Tags: limit , summation , harmonic numbers , greatest integer function , sequence

Let $(a_n)_{n\geq 1}$ be a positive real sequence given by $a_n=\sum \limits_{k=1}^n \frac{1}{k}$. Compute $$\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor$$where we denote by $\lfloor x\rfloor$ the integer part of $x$.

2010 VJIMC, Problem 1

Tags: summation , convergence

a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series $$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent? b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series $$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent. ($\mathbb N$ is the set of all positive integers.)

1996 VJIMC, Problem 2

Tags: binomial coefficient , summation , binomial , number theory , algebra

Let $\{x_n\}^\infty_{n=0}$ be the sequence such that $x_0=2$, $x_1=1$ and $x_{n+2}$ is the remainder of the number $x_{n+1}+x_n$ divided by $7$. Prove that $x_n$ is the remainder of the number $$4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k$$

2022 VTRMC, 4

Tags: series , summation , infinite sum , logarithm

Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.

1967 IMO Longlists, 44

Tags: algebra , polynomial , summation , equation , diophantine equation

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 $S(x,n) = 0.$

1996 VJIMC, Problem 2

Tags: recurrence relation , summation , binomial , algebra , sequence

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. 29

Tags: integration , summation

Prove that $$\int \limits_0^{\infty} e^{3t}\frac{4e^{4t}(3t - 1) + 2e^{2t}(15t - 17) + 18(1 - t)}{\left(1 + e^{4t} - e^{2t}\right)^2}=12\sum \limits_{k=0}^{\infty}\frac{(-1)^k}{(2k + 1)^2}-10$$

2019 India PRMO, 13

Tags: summation

Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$ ?

2006 IMO Shortlist, 2

Tags: integration , inequalities , algebra , sequence , summation , calculus

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]

2019 South Africa National Olympiad, 2

Tags: combinatorics , summation

We have a deck of $90$ cards that are numbered from $10$ to $99$ (all two-digit numbers). How many sets of three or more different cards in this deck are there such that the number on one of them is the sum of the other numbers, and those other numbers are consecutive?

1980 IMO Longlists, 16

Tags: induction , summation , counting , combinatorics

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.)

2019 Jozsef Wildt International Math Competition, W. 32

Tags: summation , inequalities , sequence

Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u_k > a_k$ and $v_k > b_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1 \leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$, then $$\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}$$where$$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$

2006 IMO Shortlist, 3

Tags: calculus , floor function , number theory , sequence , summation

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]

2022 Thailand TST, 1

Tags: algebra , floor function , summation , number theory

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?

India EGMO 2024 TST, 2

Tags: floor , algebra , summation

Given that $a_1, a_2, \dots, a_{10}$ are positive real numbers, determine the smallest possible value of \[\sum \limits_{i = 1}^{10} \left\lfloor \frac{7a_i}{a_i+a_{i+1}}\right\rfloor\] where we define $a_{11} = a_1$. [i]Proposed by Sutanay Bhattacharya[/i]

1966 IMO Longlists, 29

Tags: number theory , summation , equation

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.