Olympiad problems

1998 Tournament Of Towns, 2

For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result? ( G Galperin)

1989 Dutch Mathematical Olympiad, 3

Tags: Sum , algebra

Calculate $$\sum_{n=1}^{1989}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$

2004 German National Olympiad, 4

Tags: algebra , square roots , Sum , Integers

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

2019 Durer Math Competition Finals, 1

Tags: algebra , inequalities , Sequence , Sum

Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

2016 Thailand Mathematical Olympiad, 2

Tags: bijection , Sum , inequalities

Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

2016 Irish Math Olympiad, 3

Tags: algebra , polynomial , roots , Sum

Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?

1936 Eotvos Mathematical Competition, 1

Tags: algebra , Sum , number theory

Prove that for all positive integers $n$, $$\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+ ...+ \frac{1}{(2n - 1)2n}=\frac{1}{n + 1}\frac{1}{n + 2}+ ... +\frac{1}{2n}$$

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 $p + q > n$.

2012 Chile National Olympiad, 2

Tags: number theory , Sum , inequalities

Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$. Prove that $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

1986 Tournament Of Towns, (120) 2

Tags: geometry , perimeter , Sum , circle , square

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)

1978 Swedish Mathematical Competition, 2

Tags: Sum , number theory , Digits , algebra

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

2019 Istmo Centroamericano MO, 4

Tags: algebra , Sum

Let $x, y, z$ be nonzero real numbers such that $ x + y + z = 0$ and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1 -xyz + \frac{1}{xyz}.$$ Determine the value of the expression ' $$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$

1945 Moscow Mathematical Olympiad, 095

Tags: tangent circles , tangent , Sum , geometry

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.

2000 Tournament Of Towns, 3

Tags: least common multiple , LCM , number theory , Sum , Product , divisible , divides

The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

2001 Estonia Team Selection Test, 4

Tags: Sum , Product , algebra , combinatorics

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

2010 Korea Junior Math Olympiad, 2

Tags: combinatorics , Sum , Product , grid , board , square grid

Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?

2007 Junior Balkan Team Selection Tests - Moldova, 4

Tags: Average , Sum , combinatorics

The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?

2014 Danube Mathematical Competition, 1

Tags: number theory , Sum , primes , natural

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.

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?