Olympiad problems

1974 Chisinau City MO, 79

Tags: combinatorics , sum

There are many of the same regular triangles. At the vertices of each of them, the numbers $1, 2, 3$ are written in random order. The triangles were superimposed on one another and found the sum of the numbers that fell into each of the three corners of the stack. Could it be that in each corner the sum is equal to: a) $25$, b) $50$?

1989 Romania Team Selection Test, 1

Tags: combinatorics , sum , number theory

Let $M$ denote the set of $m\times n$ matrices with entries in the set $\{0,1,2,3,4\}$ such that in each row and each column the sum of elements is divisible by $5$. Find the cardinality of set $M$.

1986 All Soviet Union Mathematical Olympiad, 423

Tags: perfect square , table , number theory , sum , combinatorics

Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

1978 All Soviet Union Mathematical Olympiad, 252

Tags: sum , algebra

Let $a_n$ be the closest to $\sqrt n$ integer. Find the sum $$1/a_1 + 1/a_2 + ... + 1/a_{1980}$$

1977 Bundeswettbewerb Mathematik, 1

Tags: integer , algebra , number theory , sum , divisible

Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

2012 Tournament of Towns, 4

Tags: 3d geometry , cube , sum , geometry

Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.

2011 Tournament of Towns, 4

Tags: combinatorics , consecutive , sum

The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?

2016 Dutch IMO TST, 2

Tags: sum , arithmetic sequence , algebra

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

1986 Tournament Of Towns, (119) 1

Tags: number theory , digit , 2-digit , sum , difference

We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

2016 Switzerland - Final Round, 7

Tags: combinatorics , sum

There are $2n$ distinct points on a circle. The numbers $1$ through $2n$ are randomly assigned to this one points distributed. Each point is connected to exactly one other point, so that no of the resulting connecting routes intersect. If a segment connects the numbers $a$ and $b$, so we assign the value $ |a - b|$ to the segment . Show that we can choose the routes such that the sum of these values results $n^2$.

2002 Brazil National Olympiad, 1

Tags: perfect square , number theory , perfect cube , perfect power , sum , subset

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

1997 Abels Math Contest (Norwegian MO), 3a

Tags: sum , algebra , combinatorics

Each subset of $97$ out of $1997$ given real numbers has positive sum. Show that the sum of all the $1997$ numbers is positive.

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?

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 $T$.

1990 Austrian-Polish Competition, 9

Tags: sum , algebra , partition , combinatorics

$a_1, a_2, ... , a_n$ 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.

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory , reciprocal sum , sum , irreducible , divisor

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

2017 Hanoi Open Mathematics Competitions, 1

Tags: algebra , sum , polynomial

Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. The sum $|x_1| + |x_2| + |x_3|$ is (A): $4$ (B): $6$ (C): $8$ (D): $10$ (E): None of the above.

2004 Estonia Team Selection Test, 4

Tags: trigonometry , sum , rational , algebra

Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

2018 India PRMO, 25

Tags: number theory , sum , remainder , digit

Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?

1990 Swedish Mathematical Competition, 2

Tags: sum , geometric inequalities , geometry , collinear

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

1947 Putnam, A5

Tags: limit , sum , sequence

Let $a_1 , b_1 , c_1$ be positive real numbers whose sum is $1,$ and for $n=1, 2, \ldots$ we define $$a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.$$ Show that $a_n , b_n ,c_n$ approach limits as $n\to \infty$ and find those limits.

2008 Tournament Of Towns, 3

Tags: algebra , polynomial , inequalities , sum , root

A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$. Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$

2005 Singapore Senior Math Olympiad, 3

Tags: sum , subset , combinatorics

Let $S$ be a subset of $\{1,2,3,...,24\}$ with $n(S)=10$. Show that $S$ has two $2$-element subsets $\{x,y\}$ and $\{u,v\}$ such that $x+y=u+v$

2015 Dutch IMO TST, 4

Tags: maximum value , maximum , combinatorics , sum , coloring

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

1977 Bundeswettbewerb Mathematik, 2

Tags: pyramid , combinatorics , number , sum

A beetle crawls along the edges of an $n$-lateral pyramid, starting and ending at the midpoint $A$ of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)? Show that the sum of the numbers of passed vertices (over all these ways) equals $1^2 +2^2 +\ldots +n^2. $