#### NumberTheory IMO

###### back to index

Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.

Solve in positive integers $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{5}$

Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.

Prove for any positive integer $n$, the fraction $\frac{21n+4}{14n+3}$ cannot be further simplified.

If integer d is not equal to 2,5 or 13. Prove: there must exist two different elements $a$ and $b$ in set {2, 5, 3, d} such that $ab - Let$f(n)$denote the sum of the digits of$n$. Find$f(f(f(4444^{4444})))$. Prove that$\frac{5^{125}-1}{5^{25}-1}$is composite. Find all integers$a$,$b$,$c$with$1 < a < b < c$such that the number$(a-1)(b-1)(c-1)$is a divisor of$abc-1$. Prove that if positive integer$a$and$b$are such that$ab+1$divides$a^2 + b^2$. then $$\frac{a^2+b^2}{ab+1}$$ is a square number. Let sequence$g(n)$satisfy$g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$where$n\ge 1$. Show that if$n$is a prime greater than 5, then$n\mid g(n)[g(n)+1]$. Find all postive integers$(a,b,c)$such that $$ab-c,\quad bc-a,\quad ca-b$$are all powers of$2$. Proposed by Serbia Show that every integer$k > 1$has a multiple which is less than$k^4$and can be written in base 10 using at most 4 different digits. In a sports contest, there were$m$medals awarded on$n$successive days ($n > 1$). On the first day, one medal and$1/7$of the remaining$m − 1$medals were awarded. On the second day, two medals and$1/7$of the now remaining medals were awarded; and so on. On the$n^{th}$and last day, the remaining$n$medals were awarded. How many days did the contest last, and how many medals were awarded altogether? Determine all three-digit numbers$N$having the property that$N$is divisible by 11, and$\dfrac{N}{11}$is equal to the sum of the squares of the digits of$N$. Show that for any positive integer$n$, the value of$\displaystyle\sum_{k=0}^{n}2^{3k}\binom{2n+1}{2k+1}$is not a multiple of$5\$.