Olympiad problems

1999 Bosnia and Herzegovina Team Selection Test, 5

Tags: combinatorics , set , product , sum , number theory

For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that $a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$ $b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$ where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$

2014 India PRMO, 19

Tags: algebra , sum

Let $x_1,x_2,... ,x_{2014}$ be real numbers different from $1$, such that $x_1 + x_2 +...+x_{2014} = 1$ and $\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1$ What is the value of $\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}$ ?

1997 Abels Math Contest (Norwegian MO), 3a

Tags: sum , combinatorics , algebra

Each subset of $97$ out of $1997$ given real numbers has positive sum. Show that the sum of all the $1997$ numbers is positive.

1957 Moscow Mathematical Olympiad, 349

Tags: rectangle table , table , sum , combinatorics , geometry

For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.

2009 Greece JBMO TST, 4

Tags: algebra , minimum , sum , system of equations

Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.

2004 Thailand Mathematical Olympiad, 5

Tags: equation , algebra , sum , radical

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: sum , combinatorics , inequalities

We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.

2000 Austrian-Polish Competition, 1

Tags: sum , algebra , floor function , polynomial

Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$ holds for infinitely many real numbers $x$.

1998 Abels Math Contest (Norwegian MO), 3

Tags: sum , sum of powers , number theory

Let $n$ be a positive integer. (a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$. (b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.

1986 Tournament Of Towns, (119) 1

Tags: number theory , digit , 2-digit , sum , difference

We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

2000 BAMO, 1

Tags: relative prime , sum , number theory

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than 1. (Two integers are relatively prime if they share no common positive divisor other than $1$. For example, $22$ and 15 are relatively prime, and thus $37 = 22+15$ represents the number 37 in the desired way.)

2011 Tournament of Towns, 5

Tags: combinatorial geometry , angle , sum , line , combinatorics

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

2007 Greece JBMO TST, 4

Tags: radical , sum , algebra

Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$

1996 Estonia National Olympiad, 4

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

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

2012 Tournament of Towns, 3

Tags: algebra , trigonometry , sum , coordinates

Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.

1987 All Soviet Union Mathematical Olympiad, 449

Tags: composite , sum , number theory , relatively prime

Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.

1999 Tournament Of Towns, 2

Tags: prime , sum of powers , sum , number theory

Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$. (a) May it happen that $d = 2$? (b) May it happen that $d$ is prime? (V Senderov)

2009 Abels Math Contest (Norwegian MO) Final, 4b

Tags: algebra , sum , inequalities

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

1939 Moscow Mathematical Olympiad, 045

Tags: sum , geometric construction , line construction , geometry

Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.

2017 QEDMO 15th, 11

Tags: algebra , sum

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

2021 Saudi Arabia Training Tests, 39

Tags: sum , divide , divisible , number theory

Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

2013 India PRMO, 18

Tags: sum , maximum , consecutive integers , algebra

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

2000 Abels Math Contest (Norwegian MO), 2b

Tags: inequalities , algebra , sum

Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$. Show that $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.

2018 Hanoi Open Mathematics Competitions, 4

Tags: number theory , remainder , sum

A pyramid of non-negative integers is constructed as follows (a) The first row consists of only $0$, (b) The second row consists of $1$ and $1$, (c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure). Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$: A. $2$ B. $4$ C. $6$ D. $11$ E. $17$

2013 Kyiv Mathematical Festival, 2

Tags: representation , sum , perfect square , number theory

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?