ISL 1982

Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that $$\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.$$

All edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$ . Prove the inequality $$\sum_{k=1}^6 (2R_k-7)^2 \leq 54.$$

Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.

Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence $a_0, a_1, \ldots$ of descendants (i.e., $a = a_0$ and for all $n \geq 1, a_{n+1}$ is always a child of $a_n$ ). It is assumed that no-one can have infinitely many children.

Variant 1. Prove that if $a$ has infinitely many ancestors, then $a$ has an infinite descending sequence of ancestors (i.e., $a_0, a_1, \ldots$ where $a = a_0$ and $a_n$ is always a child of $a_{n+1}$ ).

Variant 2. Prove that if someone has infinitely many ancestors, then all people cannot descend from $A(dam)$ and $E(ve)$ .

Let $(a_n)_{n\geq0}$ and $(b_n)_{n \geq 0}$ be two sequences of natural numbers. Determine whether there exists a pair $(p, q)$ of natural numbers that satisfy $$p < q \text{ and } a_p \leq a_q, b_p \leq b_q.$$

Let $(u_1, \ldots, u_n)$ be an ordered $n$ tuple. For each $k, 1 \leq k \leq n$ , define $v_k=\sqrt[k]{u_1u_2 \cdots u_k}$ . Prove that $$\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.$$ ( $e$ is the base of the natural logarithm).

Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$ ? Answer the analogous question if $f$ is restricted to rationals.

Let $M$ be the set of all functions $f$ with the following properties:(i) $f$ is defined for all real numbers and takes only real values.(ii) For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$ (iii) $f(0) \neq 0.$ Determine all functions $f \in M$ such that**(a)** $f(1)=\frac 52$ ,(b) $f(1)= \sqrt 3$ .

If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.

Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows $$u_{1,1}=1, u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.$$ (the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$

Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$ .

We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.

Let $\mathfrak F$ be the family of all $k$ -element subsets of the set $\{1, 2, \ldots, 2k + 1\}$ . Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$ , the sets $A$ and $f(A)$ are disjoint.

Let $A$ and $B$ be positions of two ships $M$ and $N$ , respectively, at the moment when $N$ saw $M$ moving with constant speed $v$ following the line $Ax$ . In search of help, $N$ moves with speed $kv$ ( $k < 1$ ) along the line $By$ in order to meet $M$ as soon as possible. Denote by $C$ the point of meeting of the two ships, and set $$AB = d, \angle BAC = \alpha, 0 \leq \alpha < \frac{\pi}{2}.$$ Determine the angle $\angle ABC = \beta$ and time $t$ that $N$ needs in order to meet $M$ .

Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.

Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$ . (Here $\sec''$ means the second derivative of $\sec$ ).

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$ , show that there exists an integer $k$ ( $1 \leq k \leq n$ ) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ , the following inequality holds: $$\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.$$

Simplify $$\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.$$

Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$ . Prove that $$(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha$$ if $0 \leq \alpha \leq 1$ .

We are given $2n$ natural numbers $$1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.$$ Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$ , there are exactly $k$ numbers between the two numbers $k$ .

Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$ . If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$ , prove that for $|x| \leq 1$ ,(a) $|f(x)| \leq 5/4$ ,(b) $|g(x)| \leq 2$ .