Olympiad problems

2013 Tournament of Towns, 4

Is it true that every integer is a sum of finite number of cubes of distinct integers?

Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ? Justify your answer.

2013 India PRMO, 18

Tags: sum , maximum , consecutive integers , algebra

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

1995 Czech And Slovak Olympiad IIIA, 2

Tags: integer , sum , algebra

Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.

1996 Romania National Olympiad, 1

Tags: binomial coefficient , sum , algebra

For $n ,p \in N^*$ , $ 1 \le p \le n$, we define $$ R_n^p = \sum_{k=0}^p (p-k)^n(-1)^k C_{n+1}^k $$ Show that: $R_n^{n-p+1} =R_n^p$ .

1987 Swedish Mathematical Competition, 1

Tags: algebra , sum

Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.

2015 Ukraine Team Selection Test, 4

Tags: sum , product , number theory , modulo

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

2017 Czech And Slovak Olympiad III A, 4

Tags: combinatorics , sum , probabilistic method

For each sequence of $n$ zeros and $n$ units, we assign a number that is a number sections of the same digits in it. (For example, sequence $00111001$ has $4$ such sections $00, 111,00, 1$.) For a given $n$ we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to $(n+1){2n \choose n} $

2018 India PRMO, 22

Tags: partition , subset , sum , combinatorics , set

A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?

1991 Tournament Of Towns, (296) 3

Tags: algebra , sum , combinatorics

The numbers $x_1,x_2,x_3, ..., x_n$ satisfy the two conditions $$\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1$$ Prove that there are two numbers among them whose product is no greater than $- 1/n$. (Stolov, Kharkov)

1991 Romania Team Selection Test, 7

Tags: sum , algebra , inequalities

Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

1982 Polish MO Finals, 5

Tags: divisible , sequence , sum

Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.

2020 LIMIT Category 2, 14

Tags: sum , number theory , limit

Let $f: N \to N$ satisfy $n=\sum_{d|n} f(d), \forall n \in N$. Then sum of all possible values of $f(100)$ is?

2007 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory , digit , sum , product , sum of digits

The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

2020 Argentina National Olympiad, 5

Tags: inequalities , sum , permutation , algebra

Determine the highest possible value of: $$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$ where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$. Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.

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 ' $$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$

2013 Tournament of Towns, 4

2001 Singapore MO Open, 2

2013 India PRMO, 18

1995 Czech And Slovak Olympiad IIIA, 2

1996 Romania National Olympiad, 1

1987 Swedish Mathematical Competition, 1

2015 Ukraine Team Selection Test, 4

2017 Czech And Slovak Olympiad III A, 4

2018 India PRMO, 22

1991 Tournament Of Towns, (296) 3

1991 Romania Team Selection Test, 7

1982 Polish MO Finals, 5

2020 LIMIT Category 2, 14

2007 Junior Balkan Team Selection Tests - Moldova, 1

2020 Argentina National Olympiad, 5

2019 Istmo Centroamericano MO, 4

2014 Hanoi Open Mathematics Competitions, 15

2000 Bundeswettbewerb Mathematik, 2

2016 Hanoi Open Mathematics Competitions, 8

1999 Switzerland Team Selection Test, 2

2013 Saudi Arabia BMO TST, 8

2004 Thailand Mathematical Olympiad, 5

2022 IFYM, Sozopol, 5

2019 Durer Math Competition Finals, 11

1997 Abels Math Contest (Norwegian MO), 1