Let the sequence $\{x_n\}_{n=1}^{+\infty}$, with $|x_1|\lt 1$, be defined by
$$x_{n+1}=\frac{-x_n+\sqrt{3-3x_n^2}}2.$$
(a) What condition on $x_1$ is necessary for every term of this sequence to be positive?
(b) Is this sequence periodic? Explain your answer.
Consider the $2n-1$ consecutive natural numbers $1,2,3,\ldots,2n-1$. Cross out at least $n-1$ numbers according to these rules:
- If $a$ is crossed out, then $2a$ must be crossed out.
- If $a$ and $b$ are crossed out, then $a+b$ must be crossed out.
Which numbers should be crossed out to maximize the sum of those remaining? Explain your answer.
A tetrahedron is to be cut by three planes to obtain a parallelepiped whose three faces through one vertex lie on faces of the tetrahedron.
(a) Can the parallelepiped have volume $9/10$ of the tetrahedron's volume? Explain.
(b) Determine a common point of the three cutting planes so that the parallelepiped has volume $11/50$ of the tetrahedron's volume.
In a plane, let $ABC$ be a triangle. Determine the locus of points $M$ in the plane such that
$$MA\cdot MA'=MB\cdot MB'=MC\cdot MC',$$
where $A',B',C'$ are the perpendicular projections of $M$ onto the lines $BC,CA,AB$, respectively.
Let $f(x)$ be a nonconstant polynomial with real coefficients satisfying
$$f(x)f(2x^2)=f(2x^3+x)$$
for every $x\in\mathbb R$. Prove that $f$ has no real roots.
The children in a class stand in a circle to play a candy game. The teacher gives each child an even number of candies (every child has candies, but their numbers may differ). A child gives half of their candies to the child immediately on their right. The child who has just received candies does the same if their new number is even; if it is odd, they receive one candy from the teacher before sharing in the same way. The children continue passing candies around the circle in turn. Prove that eventually there is a situation in which, if a child gives half of their candies to the teacher instead of to their neighbor, all children have equal numbers of candies.