Olympiad problems

1977 All Soviet Union Mathematical Olympiad, 245

Tags: combinatorics , sum

Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.

1960 Poland - Second Round, 1

Tags: sum , algebra , inequalities

Prove that if the real numbers $ a $ and $ b $ are not both equal to zero, then for every natural $ n $ $$ a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0. $$

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 $480$.

2002 Estonia National Olympiad, 4

Tags: integer , number theory , sum , max

Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.

2000 Switzerland Team Selection Test, 6

Tags: inequalities , sum , algebra

Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

1957 Moscow Mathematical Olympiad, 363

Tags: number theory , sum , sequence , fibonacci

Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.

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 $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

2012 India PRMO, 16

Tags: algebra , sum , function

Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions: (a) $f(mn) =f(m)f(n)$ (b) $f(m) < f(n)$ if $m < n$ (c) $f(2) = 2$ What is the sum of $\Sigma_{k=1}^{20}f(k)$?

2018 Estonia Team Selection Test, 3

Tags: sum , algebra , inequalities , max , min

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

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 $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

2000 Bundeswettbewerb Mathematik, 1b

Tags: number theory , sum , digit

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?

2010 Saudi Arabia BMO TST, 2

Tags: algebra , sum

Evaluate the sum $$1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 1 0 + . . . - 2010$$ , where each three consecutive signs $+$ are followed by two signs $-$.

2004 Junior Tuymaada Olympiad, 2

Tags: combinatorics , sum

For which natural $ n \geq 3 $ numbers from 1 to $ n $ can be arranged by a circle so that each number does not exceed $60$ % of the sum of its two neighbors?

2016 IFYM, Sozopol, 8

Tags: power , algebra , sum

Let $a_i$, $i=1,2,…2016$, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples $x_1,x_2…x_{2016}$ of natural numbers, for which the sum $\sum_{i=1}^{2016}{a_i x_i^i}$ is a 2017-th power of a natural number.

2019 India PRMO, 30

Tags: sum , set

Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.

2014 Junior Balkan Team Selection Tests - Moldova, 1

Tags: sum , inequalities , algebra

Prove that $$\frac{2 }{2013 +1} +\frac{2^{2}}{2013^{2^{1}}+1} +\frac{2^{3}}{2013^{2^{2}}+1} + ...+ \frac{2^{2014}}{2013^{2^{2013}}+1} < \frac{1}{1006}$$

2008 Indonesia TST, 2

Tags: algebra , inequalities , sequence , sum

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

2001 Switzerland Team Selection Test, 4

Tags: number theory , sum , divisor

For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.

1993 Poland - Second Round, 5

Tags: inradius , sum , incircle , geometry

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose that the inradii of the triangles $AEF,BFD,CDE$ are all equal to $r_1$. If $r_2$ and $r$ are the inradii of triangles $DEF$ and $ABC$ respectively, prove that $r_1 +r_2 =r$.

VMEO III 2006 Shortlist, N9

Tags: number theory , sum , inequalities

Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have $$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$

1962 Polish MO Finals, 1

Tags: algebra , arithmetic progression , sum

Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then $$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$

2014 India PRMO, 19

Tags: sum , algebra

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 $\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}$ ?

1973 Chisinau City MO, 67

Tags: algebra , max , sum , product , inequalities

The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

2000 Tournament Of Towns, 4

Tags: convex polygon , lattice points , sum , combinatorial geometry , combinatorics

Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon. (G Galperin)

1999 North Macedonia National Olympiad, 1

Tags: sum , algebra , positive , combinatorics

In a set of $21$ real numbers, the sum of any $10$ numbers is less than the sum of the remaining $11$ numbers. Prove that all the numbers are positive.