Let $a$ be a nonnegative real number. Define $u_1=3$ and
$$u_{n+1}=\frac12u_n+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3}\qquad(n\ge1).$$
a) For $a=0$, prove that the sequence has a finite limit and find it.
b) For every $a\in[0,1]$, prove that the sequence has a finite limit.
Let $K$ be a positive integer. Find the number of natural numbers $n\le10^K$ satisfying both:
i) $n$ is divisible by 3;
ii) every digit of its decimal representation belongs to $\{2,0,1,5\}$.
Let $(O)$ be a circle and $B,C$ fixed points on it such that $BC$ is not a diameter. A point $A$ varies on $(O)$ so that triangle $ABC$ is acute. Let $E,F$ be the feet of the altitudes from $B,C$, respectively. Let $(I)$ be a variable circle through $E,F$, with center $I$.
a) Suppose $(I)$ is tangent to $BC$ at $D$. Prove $\dfrac{DB}{DC}=\sqrt{\dfrac{\cot B}{\cot C}}$.
b) Suppose $(I)$ meets segment $BC$ at $M,N$. Let $H$ be the orthocenter of $ABC$, and $P,Q$ the intersections of $(I)$ with the circumcircle of $HBC$. A circle $(K)$ passes through $P,Q$ and is tangent to $(O)$ at $T$, where $T$ and $A$ are on the same side of $PQ$. Prove that the internal bisector of $\angle MTN$ always passes through a fixed point.
Define polynomials by
$$f_0(x)=2,\quad f_1(x)=3x,\quad f_n(x)=3xf_{n-1}(x)+(1-x-2x^2)f_{n-2}(x)\quad(n\ge2).$$
Find all positive integers $n$ for which $f_n(x)$ is divisible by $x^3-x^2+x$.
For positive integers $a,n$, consider
$$a^2x+6ay+36z=n,$$
where $x,y,z$ are natural numbers.
a) Find all $a$ such that for every $n\ge250$ the equation has a solution in natural numbers $(x,y,z)$.
b) Suppose $a\gt 1$ and $\gcd(a,6)=1$. Find, in terms of $a$, the largest $n$ for which the equation has no solution $(x,y,z)$.
There are $m$ girls and $n$ boys, $m,n\ge2$, taking part in a duet festival. Each session presents one program. A program consists of male–female duets, with each girl–boy pair singing at most one song and every student singing at least one song. Two programs are different if some girl–boy pair sings in one program but not the other. The festival ends only after every possible different program has been performed exactly once.
a) A program is called dependent on student $X$ if canceling all duets involving $X$ leaves at least one other student singing no song in that program. Prove that among all programs dependent on $X$, the number with an odd number of songs equals the number with an even number of songs.
b) Prove that the organizers can order the sessions so that any two consecutive sessions have different parities of their numbers of songs.