Olympiad problems

2016 AMC 8, 11

Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$ $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }12$

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , prime divisors , Fractions , harmonic , numbers

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , numbers , circle

The numbers from $1$ through $100$ are written in some order on a circle. We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.

Bangladesh Mathematical Olympiad 2020 Final, #9

Tags: numbers , Numberteory , number theory

You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer[b] k[/b], in which piles contain at least[b] k [/b]coins, take away exact[b] k[/b] coins from these piles. Find the [b]minimum number of turns[/b] you need to take way all of these coins?

2017 VTRMC, 1

Tags: numbers

Determine the number of real solutions to the equation $\sqrt{2 -x^2} = \sqrt[3]{3 -x^3}.$

1999 Estonia National Olympiad, 5

Tags: circle , numbers , consecutive , Sum , number theory , combinatorics

The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least $15$, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.

Bangladesh Mathematical Olympiad 2020 Final, #7

Tags: number theory , numbers

Tiham is trying to find [b]6[/b] digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the [b]3[/b] digit number$ PQR$, and the [b]3[/b] digit number $STU$ is divisible by [b]37[/b]. How many such numbers Tiham can find?

2009 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , circle , numbers

On circle there are $2009$ positive integers which sum is $7036$. Show that it is possible to find two pairs of neighboring numbers such that sum of both pairs is greater or equal to $8$

2020 Bosnia and Herzegovina Junior BMO TST, 2

Tags: board , combinatorics , numbers

A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square. Such a board is called [i] interesting[/i] if the following conditions hold: $\circ$ In all unit squares below the main diagonal, the number $0$ is written; $\circ$ Positive integers are written in all other unit squares. $\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers are actually the numbers $1,2,...,2n$ (not necessarily in that order). $a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board. $b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.

2016 Harvard-MIT Mathematics Tournament, 8

Tags: numbers , number theory

For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by \[ W(n,k) = \begin{cases} n^n & k = 0 \\ W(W(n,k-1), k-1) & k > 0. \end{cases} \] Find the last three digits in the decimal representation of $W(555,2)$.

2015 IMC, 2

Tags: numbers , college contests , inequalities , IMC2015

For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) =8$. Prove that \[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\] When does equality hold? (Proposed by Stephan Wagner, Stellenbosch University)