Olympiad problems

1984 Swedish Mathematical Competition, 6

Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$. Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.

2000 BAMO, 1

Tags: number theory , relative prime , sum

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.)

2018 Lusophon Mathematical Olympiad, 1

Tags: sum , algebra

Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$.

1991 All Soviet Union Mathematical Olympiad, 545

Tags: game , game strategy , combinatorics , sum , difference

The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$

1953 Poland - Second Round, 2

Tags: algebra , sum

The board was placed $$ \begin{array}{rcl} 1 & = & 1 \\ 2 + 3 + 4 & = & 1 + 8 \\ 5 + 6 + 7 + 8 + 9 & = & 8 + 27\\ 10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64\\ & \ldots & \end{array}$$ Write such a formula for the $ n $-th row of the array that, with the substitutions $ n = 1, 2, 3, 4 $, would give the above four lines of the array and would be true for every natural $ n $.

2004 BAMO, 4

Tags: algebra , sum , inequalities

Suppose one is given $n$ real numbers, not all zero, but such that their sum is zero. Prove that one can label these numbers $a_1, a_2, ..., a_n$ in such a manner that $a_1a_2 + a_2a_3 +...+a_{n-1}a_n + a_na_1 < 0$.

2004 Bosnia and Herzegovina Junior BMO TST, 3

Tags: algebra , sum , fraction

Let $a, b, c, d$ be reals such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7$ and $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12$. Find the value of $w =\frac{a}{b}+\frac{c}{d}$ .

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2012 Tournament of Towns, 3

Tags: trigonometry , sum , coordinates , algebra

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 $\frac{a}{b}$.

2003 Junior Tuymaada Olympiad, 4

Tags: algebra , sum , maximum value , maximum , inequalities

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$

2006 Tournament of Towns, 1

Tags: combinatorics , sum

Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. With the new pair of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)

1981 Spain Mathematical Olympiad, 1

Tags: sum , algebra

Calculate the sum of $n$ addends $$7 + 77 + 777 +...+ 7... 7.$$

1985 All Soviet Union Mathematical Olympiad, 401

Tags: triangle table , combinatorics , minimum , sum , pascal triangle

In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$? j h i e f g a b c d

1987 All Soviet Union Mathematical Olympiad, 453

Tags: combinatorics , square table , table , sum

Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.

2017 QEDMO 15th, 11

Tags: sum , algebra

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

1993 Czech And Slovak Olympiad IIIA, 4

Tags: sum of powers , digit , sum , number theory , sequence

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

1988 All Soviet Union Mathematical Olympiad, 482

Tags: combinatorics , sum , algebra

Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.

1999 Tournament Of Towns, 2

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

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 Chile National Olympiad, 3

Tags: number theory , sum

Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?

2006 MOP Homework, 5

Tags: inequalities , algebra , sum , maximal , max

Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?

2014 Gulf Math Olympiad, 4

Tags: combinatorics , sum , chessboard

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

2013 Flanders Math Olympiad, 1

Tags: number theory , digit , divisible , sum

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

2011 Indonesia TST, 1

Tags: inequalities , sum , algebra

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

1996 Mexico National Olympiad, 5

Tags: combinatorics , square , perfect cube , sum

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.

2019 Argentina National Olympiad, 2

Tags: algebra , sum , inequalities

Let $n\geq1$ be an integer. We have two sequences, each of $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ and $b_1,b_2,\ldots ,b_n$ such that $a_1+a_2+\ldots +a_n=1$ and $ b_1+b_2+\ldots +b_n=1$. Find the smallest possible value that the sum can take $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.$$