Školjka

%V0 Which natural numbers can be expressed as the difference of squares of two integers?

%V0 $(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

%V0 $(NET 3)$ 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?

%V0 $(MON 4)$ 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 $3.$

%V0 $(HUN 1)$ 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 $a^2 +ab+ b^2$

%V0 $(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$