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IMO Shortlist 1969 problem 24

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ , where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$ . Show that if $P(x)$ is divisible by $m$ , then $a_0 \cdot k!$ is a multiple of $m$ . Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$ , then a polynomial $P(x)$ with leading term $ax^k$ can be found that is divisible by $m.$

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IMO Shortlist 1969 problem 34

$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$ , then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

IMO Shortlist 1969 problem 43

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$ , then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

IMO Shortlist 1969 problem 48

$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying
$x_1 +x_2 +x_3 +x_4 +x_5 = 1000,$
$x_1 -x_2 +x_3 -x_4 +x_5 > 0,$
$x_1 +x_2 -x_3 +x_4 -x_5 > 0,$
$-x_1 +x_2 +x_3 -x_4 +x_5 > 0,$
$x_1 -x_2 +x_3 +x_4 -x_5 > 0,$
$-x_1 +x_2 -x_3 +x_4 +x_5 > 0$
$(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$
$(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?