Olympiad problems

2022 China Northern MO, 2

(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

1956 Moscow Mathematical Olympiad, 323

Tags: number theory , Divide , divisor

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

1955 Moscow Mathematical Olympiad, 292

Tags: number theory , Divide

Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$. Prove that if $n > 3$, then $6$ divides $ab$.

2010 Thailand Mathematical Olympiad, 6

Tags: primes , number theory , Divide

Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

1999 Singapore Team Selection Test, 3

Tags: number theory , prime numbers , Divide

Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$

1983 Tournament Of Towns, (038) A5

Tags: number theory , divisible , Divide , combinatorics

Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)

1952 Moscow Mathematical Olympiad, 232

Tags: polynomial , Divide , algebra

Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

1955 Moscow Mathematical Olympiad, 290

Tags: number theory , divisible , Divide

Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

1945 Moscow Mathematical Olympiad, 094

Tags: geometry , Impossible , Divide , congruent triangles , combinatorial geometry

Prove that it is impossible to divide a scalene triangle into two equal triangles.

1963 German National Olympiad, 1

Tags: number theory , remainder , Divide

a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

1986 Tournament Of Towns, (129) 4

Tags: factorial , number theory , Divide , divisible , divisor

We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

2008 Mathcenter Contest, 6

Tags: number theory , Divide

For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$. [i](tomoyo-jung)[/i]

2010 Saudi Arabia BMO TST, 3

Tags: number theory , Divide , divisible

How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

1954 Moscow Mathematical Olympiad, 267

Tags: polynomial , divisible , Divide , algebra

Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

1986 Tournament Of Towns, (108) 2

Tags: number theory , maximum , Digits , Divide

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

1947 Moscow Mathematical Olympiad, 136

Tags: 13-gon , parallelogram , Divide , convex , combinatorics

Prove that no convex $13$-gon can be cut into parallelograms.