Olympiad problems

2017 Irish Math Olympiad, 5

Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies $$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$. (a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is $2017$-powerful.

1992 Romania Team Selection Test, 4

Tags: sum , algebra , inequalities

Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$. If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.

1992 Chile National Olympiad, 7

Tags: magic square , combinatorics , sum , board

$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$. $\bullet$ Construct or describe the construction of this magic square.

2011 Tournament of Towns, 2

Tags: algebra , max , sum , combinatorics

$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$. [hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]

2021 Austrian Junior Regional Competition, 1

Tags: algebra , sum , cool , easy

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

2021 Polish Junior MO First Round, 5

Tags: number theory , sum , product

Are there four positive integers whose sum is $2^{1002}$ and product is $5^{1002}$? Justify your answer.

2002 Spain Mathematical Olympiad, Problem 4

Tags: number theory , sum

Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.

1988 Swedish Mathematical Competition, 3

Tags: inequalities , sum , algebra

Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$. Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$.

2009 Danube Mathematical Competition, 2

Tags: sum , consecutive , number theory

Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .

2007 Singapore Junior Math Olympiad, 4

Tags: number theory , greatest common divisor , least common multiple , sum , difference , product

The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.

2000 Junior Balkan Team Selection Tests - Moldova, 1

Tags: algebra , sum

Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.

2016 India PRMO, 12

Tags: sum , radical , floor function , algebra , inequalities

Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$. You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.

1983 Swedish Mathematical Competition, 1

Tags: sum , algebra

The positive integers are grouped as follows: $1, 2+3, 4+5+6, 7+8+9+10,\dots$. Find the value of the $n$-th sum.

2007 Thailand Mathematical Olympiad, 8

Tags: algebra , sum , polynomial

Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

2004 VJIMC, Problem 2

Tags: summation , sum , real analysis

Evaluate the sum $$\sum_{n=0}^\infty\operatorname{arctan}\left(\frac1{1+n+n^2}\right).$$

1999 Cono Sur Olympiad, 3

Tags: combinatorics , sum , coloring

There are $1999$ balls in a row, some are red and some are blue (it could be all red or all blue). Under every ball we write a number equal to the sum of the amount of red balls in the right of this ball plus the sum of the amount of the blue balls that are in the left of this ball. In the sequence of numbers that we get with this balls we have exactly three numbers that appears an odd number of times, which numbers could these three be?

2009 Kyiv Mathematical Festival, 5

Tags: geometry , sum , distance , area of triangle

Assume that a triangle $ABC$ satisfies the following property: For any point from the triangle, the sum of distances from $D$ to the lines $AB,BC$ and $CA$ is less than $1$. Prove that the area of the triangle is less than or equal to $\frac{1}{\sqrt3}$

2015 NIMO Summer Contest, 2

Tags: sum

On a 30 question test, Question 1 is worth one point, Question 2 is worth two points, and so on up to Question 30. David takes the test and afterward finds out he answered nine of the questions incorrectly. However, he was not told which nine were incorrect. What is the highest possible score he could have attained? [i] Proposed by David Altizio [/i]

2012 IMAR Test, 2

Tags: number theory , sum , coprime

Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.

2015 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , positive integer , remainder , sum

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

1995 Czech and Slovak Match, 4

Tags: minimum value , algebra , sum

For each real number $p > 1$, find the minimum possible value of the sum $x+y$, where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$.

2012 Korea Junior Math Olympiad, 7

Tags: maximum , positive real , sum , algebra , inequalities

If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)