Olympiad problems

1996 All-Russian Olympiad Regional Round, 9.5

Find all natural numbers that have exactly six divisors whose sum is $3500$.

1974 Poland - Second Round, 2

Tags: algebra , sum , inequalities

Prove that for every $ n = 2, 3, \ldots $ and any real numbers $ t_1, t_2, \ldots, t_n $, $ s_1, s_2, \ldots, s_n $, if $$ \sum_{i=1}^n t_i = 0, \text{ to } \sum_{i=1}^n\sum_{j=1}^n t_it_j |s_i-s_j| \leq 0.$$

2013 Tournament of Towns, 2

Tags: sum , algebra

Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.

1986 Tournament Of Towns, (120) 2

Tags: geometry , perimeter , sum , square , circles

Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that (a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$. (b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$. ( V . V . Proizvolov , Moscow)

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 $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

2016 Czech And Slovak Olympiad III A, 1

Tags: number theory , sum , fraction , integer

Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.

2020 Nordic, 3

Tags: trisector , sum , area of a triangle , geometry , diagonal , area

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

2000 Singapore Senior Math Olympiad, 3

Tags: number theory , inequalities , sum , lcm

Let $n_1,n_2,n_3,...,n_{2000}$ be $2000$ positive integers satisfying $n_1<n_2<n_3<...<n_{2000}$. Prove that $$\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}$$ where $[a, b]$ denotes the least common multiple of $a$ and $b$.

1981 Tournament Of Towns, (013) 3

Tags: number theory , decimal representation , sum

Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$. (E Turkevich)

2018 Dutch IMO TST, 3

Tags: number theory , sum , floor function , perfect square

Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is defined as follows: we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.

1986 All Soviet Union Mathematical Olympiad, 437

Tags: number theory , integer , reciprocal , sum

Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.

2016 JBMO Shortlist, 2

Tags: combinatorics , sum , prime

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

2011 German National Olympiad, 4

Tags: point , angle , maximal , sum , set , geometry

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

2013 India PRMO, 20

Tags: number theory , sum

What is the sum (in base $10$) of all the natural numbers less than $64$ which have exactly three ones in their base $2$ representation?

2014 Danube Mathematical Competition, 1

Tags: number theory , sum , natural , prime

Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

2002 India IMO Training Camp, 16

Tags: sum , modular arithmetic , number theory , additive number theory

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

2008 Korea Junior Math Olympiad, 6

Tags: number theory , sum of powers , sum , divisible , divisor

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.

1999 Switzerland Team Selection Test, 8

Tags: sum , system of equations , algebra

Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with $$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$

2005 Thailand Mathematical Olympiad, 7

Tags: combinatorics , sum

How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?

1997 Argentina National Olympiad, 3

Tags: algebra , sum , inequalities

Let $x_1,x_2,x_3,\ldots ,x_{100}$ be one hundred real numbers greater than or equal to $0$ and less than or equal to $1$. Find the maximum possible value of the sum$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$

1999 Poland - Second Round, 6

Tags: factorial , number theory , sum

Suppose that $a_1,a_2,...,a_n$ are integers such that $a_1 +2^ia_2 +3^ia_3 +...+n^ia_n = 0$ for $i = 1,2,...,k -1$, where $k \ge 2$ is a given integer. Prove that $a_1+2^ka_2+3^ka_3+...+n^ka_n$ is divisible by $k!$.

2018 Estonia Team Selection Test, 3

Tags: sum , algebra , inequalities , max , min

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

1998 Belarus Team Selection Test, 1

Tags: inequalities , number theory , sum , divisor

Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$). Prove that $(S(n))^3 <n^4$.

1985 Tournament Of Towns, (102) 6

Tags: sequence , sum , integer part , recurrence relation , algebra

The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$ {A. Andjans, Riga)

2019 Tournament Of Towns, 4

Tags: coloring , sum , product , combinatorics

Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result? (Ilya Bogdanov)