Olympiad problems

2019 Dutch Mathematical Olympiad, 4

The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$. Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.

2013 Tournament of Towns, 2

Tags: consecutive , number theory , Sum , Product

A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.

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 $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

1990 Tournament Of Towns, (243) 1

Tags: algebra , Sum

For every natural number $n$ prove that $$\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left( \frac{1}{n-1} + \frac12 \right)^2+ \left( \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)$$ (S. Manukian, Yerevan)

2000 Abels Math Contest (Norwegian MO), 2a

Tags: algebra , Sum

Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.

2008 Singapore Junior Math Olympiad, 4

Tags: Sum , number theory , primes

Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?

2012 Oral Moscow Geometry Olympiad, 5

Tags: minimum , Sum , distance , geometry , circle

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

2013 Saudi Arabia BMO TST, 3

Tags: algebra , max , Sum , Product

Let $T$ be a real number satisfying the property: For any nonnegative real numbers $a, b, c,d, e$ with their sum equal to $1$, it is possible to arrange them around a circle such that the products of any two neighboring numbers are no greater than $T$. Determine the minimum value of $T$.

2003 Junior Balkan Team Selection Tests - Moldova, 8

Tags: Sum , combinatorics , combinatorial geometry

In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

2018 Hanoi Open Mathematics Competitions, 7

Tags: power of 2 , Sum , max , number theory

Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard?

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.

2013 Flanders Math Olympiad, 1

Tags: number theory , Digits , 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 $13$.

2003 Singapore Senior Math Olympiad, 3

Tags: algebra , Sum , Product

(i) Find a formula for $S_n = -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^n n^2 \times (n + 1)$ in terms of the positive integer $n$. Justify your answer. (As an example, one has $1 + 2 + 3 +...+n = \frac{n(n+1)}{2}$) (ii) Using your formula in (i), find the value of $ -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^{100} 100^2 \times (100 + 1)$

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.

ICMC 3, 2

Tags: college contests , complex numbers , Sum , roots of unity

Find integers $a$ and $b$ such that \[a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.\] [i]proposed by the ICMC Problem Committee[/i]

1988 Tournament Of Towns, (186) 3

Tags: number theory , divisible , Sum , divides

Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

1976 Spain Mathematical Olympiad, 2

Tags: vector , Sum , algebra

Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.

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 $1987$.

1996 Estonia National Olympiad, 4

Tags: number theory , divisible , divides , Sum of powers , Sum

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

2004 Thailand Mathematical Olympiad, 9

Tags: algebra , Sum , factorial

Compute the sum $$\sum_{k=0}^{n}\frac{(2n)!}{k!^2(n-k)!^2}.$$

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.

1983 All Soviet Union Mathematical Olympiad, 362

Tags: infinite board , Squares , Sum , combinatorial geometry , combinatorics

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?

1990 Austrian-Polish Competition, 9

Tags: Sum , algebra , combinatorics , partition

$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.

1997 Tuymaada Olympiad, 3

Tags: combinatorics , Coloring , Sum

Is it possible to paint all natural numbers in $6$ colors, for each one color to be used and the sum of any five numbers of different color to be painted in the sixth color?

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 $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.