Olympiad problems

1954 Putnam, B2

Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$. Denote by $s_k, S_k$ respectively the $k$-th partial sums of both series. Prove that $$ \!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$ \text{ii}) \; S\ne s.$$

1989 Bulgaria National Olympiad, Problem 2

Tags: series , floor function , limit , sequence , algebra

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

2015 IMC, 6

Tags: inequalities , series

Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.$$ Proposed by Ivan Krijan, University of Zagreb

1954 Miklós Schweitzer, 2

Tags: sequence , series

[b]2.[/b] Show that the series $\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}$ is convergent for every positive integer N and any real numbers a and b. [b](S. 25)[/b]

1941 Putnam, B2

Tags: convergence , series

Find (i) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}$. (ii) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}$. (iii) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$.

2002 Croatia National Olympiad, Problem 1

Tags: series , algebra

For each $x$ with $|x|<1$, compute the sum of the series $$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$

2025 SEEMOUS, P4

Tags: sequence , real analysis , series

Let $(a_n)_{n\geq 1}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $\sum_{n=1}^{\infty}\frac{a_n}{n}$ is convergent if and only if the sequence $(a_n\ln n)_{n\geq 1}$ is bounded and $\sum_{n=1}^{\infty} (a_n-a_{n+1})\ln n$ is convergent.

2003 Gheorghe Vranceanu, 3

Tags: function , convergence , series , floor function , real analysis

Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $ [b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $ [b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.

2018 Korea USCM, 8

Tags: sequence , series , inequalities

Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and $$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$ for every $n\geq 1$. (1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$. (2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.

KoMaL A Problems 2021/2022, A. 810

Tags: series , algebra , combinatorics

For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$

2016 Korea USCM, 2

Tags: sequence , series , real analysis

Suppose $\{a_n\}$ is a decreasing sequence of reals and $\lim\limits_{n\to\infty} a_n = 0$. If $S_{2^k} - 2^k a_{2^k} \leq 1$ for any positive integer $k$, show that $$\sum_{n=1}^{\infty} a_n \leq 1$$ (At here, $S_m = \sum_{n=1}^m a_n$ is a partial sum of $\{a_n\}$.)

1953 Putnam, B1

Tags: convergence , series

Is the infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ convergent?

1960 Putnam, B2

Tags: infinite series , series

Evaluate the double series $$\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3k -j -(k+j)^{2}}.$$

1985 Traian Lălescu, 1.2

Tags: series , limit , calculus , real analysis

Calculate $ \sum_{i=2}^{\infty}\frac{i^2-2}{i!} . $

1961 Putnam, A3

Tags: limit , series

Evaluate $$\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.$$

1965 Spain Mathematical Olympiad, 8

Tags: series , geometry

Let be $\gamma_1$ a circumference of radius $r$ and $P$ an exterior point that is at distance $a$ from the centre of $\gamma_1$. We build two tangent lines $r,s$ to $\gamma_1$ from $P$ and $\gamma_2$ is constructed as a smaller circumference, tangent to both lines and, also, tangent to $\gamma_1$. We construct inductively $\gamma_{n+1}$ as a tangent circumference to $\gamma_{n}$ and, also, tangent to $r$ and $s$. Determine: a) The radius of $\gamma_2$. b) The radius of $\gamma_n$. c) The sum of the lengths of $\gamma_1, \gamma_2, \gamma_3, ...$.

1960 Putnam, B6

Tags: p-adic , series

Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that $$\sum_{n=1}^{\infty} b_n$$ converges.

2024 CIIM, 6

Tags: real analysis , series , fractional part , convergence , ana

Given a real number $x$, define the series \[ S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\}, \] where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.

2024 Mexican University Math Olympiad, 4

Tags: limit , linear algebra , series , matrices

Given $ b > 0 $, consider the following matrix: \[ B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix} \] Denote by $ e_i $ the top left entry of $ B^i $. Prove that the following limit exists and calculate its value: \[ \lim_{i \to \infty} \sqrt[i]{e_i}. \]

2022 Brazil Undergrad MO, 4

Tags: real analysis , series

Let $\alpha, c > 0$, define $x_1 = c$ and let $x_{n + 1} = x_n e^{-x_n^\alpha}$ for $n \geq 1$. For which values of $\beta$ does $\sum_{i = 1}^{\infty} x_n^\beta$ converge?

2012 Centers of Excellency of Suceava, 3

Tags: real analysis , factorial , series , definite integral , integral sequence

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

2003 Miklós Schweitzer, 8

Tags: series , function

Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)? (translated by L. Erdős)

1952 Putnam, B5

Tags: convergence , series

If the terms of a sequence $a_{1}, a_{2}, \ldots$ are monotonic, and if $\sum_{n=1}^{\infty} a_n$ converges, show that $\sum_{n=1}^{\infty} n(a_{n} -a_{n+1 })$ converges.

2020 AMC 12/AHSME, 22

Tags: series

Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that \[ (2 + i)^n = a_n + b_ni \] for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] $\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47$

2022 VTRMC, 4

Tags: logarithm , series , summation , infinite sum

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