Olympiad problems

1997 Poland - Second Round, 5

Tags: probability , sums , Dice , remainder , number theory , combinatorics

We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

2001 IMO Shortlist, 6

Tags: modular arithmetic , number theory , Additive Number Theory , sums , IMO Shortlist

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

2020 Malaysia IMONST 1, 1

Tags: arithmetic , sums , IMONST

Find the value of \[+1+2+3-4-5-6+7+8+9-10-11-12+\cdots -2020,\] where the sign alternates between $+$ and $-$ after every three numbers.

2015 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , positive integer , sums , remainder

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

2011 Tournament of Towns, 2

Tags: sums , combinatorics , algebra , max

$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$. [hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]

2014 AIME Problems, 7

Tags: trigonometry , logarithms , AMC , sums , AIME , AIME II , 2014 AIME II

Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]

2014 Greece JBMO TST, 4

Tags: combinatorics , set theory , Sets , Subsets , partition , sums , Sum

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$

2002 India IMO Training Camp, 16

Tags: modular arithmetic , number theory , Additive Number Theory , sums , IMO Shortlist

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

2002 Spain Mathematical Olympiad, Problem 4

Tags: number theory , sums

Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.

2015 IFYM, Sozopol, 7

Tags: number theory , set theory , sums

Determine the greatest natural number $n$, such that for each set $S$ of 2015 different integers there exist 2 subsets of $S$ (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by $n$.

2016 Dutch IMO TST, 2

Tags: arithmetic sequence , algebra , sums

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).

1976 Euclid, 2

Tags: arithmetic sequence , sums

Source: 1976 Euclid Part A Problem 2 ----- The sum of the series $2+5+8+11+14+...+50$ equals $\textbf{(A) } 90 \qquad \textbf{(B) } 425 \qquad \textbf{(C) } 416 \qquad \textbf{(D) } 442 \qquad \textbf{(E) } 495$

2002 India IMO Training Camp, 16

Tags: modular arithmetic , number theory , Additive Number Theory , sums , IMO Shortlist

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

2016 Dutch IMO TST, 2

Tags: arithmetic sequence , algebra , sums

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).

2019 India PRMO, 30

Tags: Sets , sums

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 Contests, 4

Tags: combinatorics , set theory , Sets , Subsets , partition , sums , Sum

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$