Olympiad problems

1998 Czech And Slovak Olympiad IIIA, 2

Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$

1971 Dutch Mathematical Olympiad, 4

Tags: number theory , Sum

For every positive integer $n$ there exist unambiguously determined non-negative integers $a(n)$ and $b(n)$ such that $$n = 2^{a(n)}(2b(n)+1),$$ For positive integer $k$ we define $S(k)$ by: $$a(1) + a(2) + ... + a(2^k) = S(k)$$ Express $S(k)$ in terms of $k$.

2002 Mexico National Olympiad, 5

Tags: number theory , Divisors , maximum , Sum

A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.

2017 QEDMO 15th, 11

Tags: Sum , algebra

Calculate $$\frac{(2^1+3^1)(2^2+3^2)(2^4+3^4)(2^8+3^8)...(2^{2048}+3^{2048})+2^{4096}}{3^{4096}}$$

1983 Poland - Second Round, 4

Tags: number theory , divisor , Sum

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

1996 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , Sum , board

The cells of an $n \times n$ board are labelled with the numbers $1$ through $n^2$ in the usual way. Let $n$ of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.

1974 Swedish Mathematical Competition, 1

Tags: Sequence , algebra , Sum

Let $a_n = 2^{n-1}$ for $n > 0$. Let \[ b_n = \sum\limits_{r+s \leq n} a_ra_s \] Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.

2016 All-Russian Olympiad, 5

Tags: number theory , Sum

Using each of the digits $1,2,3,\ldots ,8,9$ exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in $n$ zeroes,where $n$ is a non-negative integer.Determine the maximum possible value of $n$.

2002 Brazil National Olympiad, 1

Tags: number theory , Perfect Squares , perfect cube , Perfect Powers , Subset , Sum

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

2000 Bundeswettbewerb Mathematik, 1b

Tags: number theory , Sum , Digits

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?

2014 Contests, 4

Tags: combinatorics , set theory , Sets , Subsets , partition , sums , Sum

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$

2000 Switzerland Team Selection Test, 8

Tags: algebra , Sum

Let $f(x) = \frac{4^x}{4^x+2}$ for $x > 0$. Evaluate $\sum_{k=1}^{1920}f\left(\frac{k}{1921}\right)$

1992 Austrian-Polish Competition, 1

Tags: Sum , positive , Divisors , number theory , Product , Even

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

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 $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.

2011 Belarus Team Selection Test, 1

Tags: number theory , Digits , Sum , divisible

Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$. I.Kozlov

1999 Czech And Slovak Olympiad IIIA, 3

Tags: median , Sum , ratio , geometry

Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.

2014 Korea Junior Math Olympiad, 2

Tags: algebra , Sum , inequalities , Inequality

Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$. Let $s = a_1 + a_3 +...+ a_{2n-1}$, $t = a_2 + a_4 + ... + a_{2n}$, and $x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$). Prove that $$\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}$$

1991 ITAMO, 3

Tags: number theory , Sum , Sum of Squares

We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)

1985 Poland - Second Round, 2

Tags: number theory , Sum , Divisors

Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.

1984 Tournament Of Towns, (077) 2

Tags: number theory , combinatorics , remainder , Sum , permutations

A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers $b_1=a_1$ $b_2= a_1 + a_2$ $b_3=a_1 + a_2 + a_3$ ... $b_{100}=a_1 + a_2 + ...+a_{100}$ Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different . ( L . D . Kurlyandchik , Leningrad)

1999 Estonia National Olympiad, 2

Tags: Sum , algebra

Find the value of the expression $$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$ assuming $f(x) =\frac{x^2}{1 + x^2}$ .

2022 Indonesia TST, C

Tags: combinatorics , Sum , equal , choose

Five numbers are chosen from $\{1, 2, \ldots, n\}$. Determine the largest $n$ such that we can always pick some of the 5 chosen numbers so that they can be made into two groups whose numbers have the same sum (a group may contain only one number).

2015 Saudi Arabia IMO TST, 1

Tags: combinatorics , Sum

Let $S$ be a positive integer divisible by all the integers $1, 2,...,2015$ and $a_1, a_2,..., a_k$ numbers in $\{1, 2,...,2015\}$ such that $2S \le a_1 + a_2 + ... + a_k$. Prove that we can select from $a_1, a_2,..., a_k$ some numbers so that the sum of these selected numbers is equal to $S$. Lê Anh Vinh

1984 Polish MO Finals, 1

Tags: function , algebra , Sum , composition

Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.

2004 Thailand Mathematical Olympiad, 19

Tags: algebra , Sum , max , inequalities

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$