a.) For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? b.) For which $n>2$ is there exactly one set having this property?

Click for solution Let $k = \prod p_{i}^{e_{i}}$ be the largest element of the set. Then $k$ divides the least common multiple of the other elements of the set iff the set has cardinality of at least $\max \{ p_{i}^{e_{i}}\}+1$, since for any of the $p_{i}^{e_{i}}$, we must go down at least to $k-p_{i}^{e_{i}}$ to obtain another multiple of $p_{i}^{e_{i}}$. In particular, there is no set of cardinality $3$ satisfying our conditions, because each number larger than or equal to $3$ must be divisible by a number that is larger than two and is a power of a prime. For $n > 3$, we may let $k = \operatorname{lcm}[n-1, n-2] = (n-1)(n-2)$, since all the $p_{i}^{e_{i}}$ must clearly be less than $n$ and this product must also be larger than $ n$ if $n$ is at least $4$. For $n > 4$, we may also let $k = \operatorname{lcm}[n-2, n-3] = (n-2)(n-3)$, for the same reasons. However, for $ n = 4$, this does not work, and indeed no set works other than $\{ 3,4,5,6 \}$. To prove this, we simply note that for any integer not equal to $6$ and larger than $4$ must have some power-of-a-prime factor larger than $3$. Q.E.D.

A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$

Find the minimum value of \[\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\] subject to the constraints (i) $a, b, c, d, e, f, g \geq 0,$ (ii)$ a + b + c + d + e + f + g = 1.$

Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.

Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than $1$. Let \[P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.\] Let also $P_0 = Q_0$ and $P_1 = Q_1$. Prove that $P(z) \equiv Q(z).$

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

A sequence $(a_n)$ is defined by means of the recursion \[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\] Find an explicit formula for $a_n.$

Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).

On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$, respectively, are given. Prove that \[a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.\]

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Let $P$ be a polynomial of degree $n$ satisfying \[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\] Determine $P(n + 1).$

Prove that a convex pentagon (a five-sided polygon) $ABCDE$ with equal sides and for which the interior angles satisfy the condition $\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E$ is a regular pentagon.

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$: \[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\] Describe, with proof, the behavior of $u_n$ as $n \to \infty.$

Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

Click for solution Let A, B, C be the centers of the 3 congruent circles (A), (B), (C) intersecting at the point O. Since the point O is equidistant from the centers A, B, C, it is the circumcenter of the triangle $\triangle ABC$. Let a, b, c be the common external tangents of the circle pairs (B), (C); (C), (A); (A), (B), neither of them intersecting the remaining circle and let the lines a, b, c intersect at points $A' \equiv b \cap c,\ B' \equiv c \cap a,\ C' \equiv a \cap b$, forming a triangle $\triangle A'B'C'$. Let O' be the circumcenter of this new triangle. Since the common external tangent of 2 congruent circles is parallel to their center line, $a \equiv B'C' \parallel BC,\ b \equiv C'A' \parallel CA,\ c \equiv A'B' \parallel AB$. Thus the triangles $\triangle A'B'C' \sim \triangle ABC$ are centrally similar, having the corresponding sides parallel. The lines A'A, B'B, C'C connecting the corresponding vertices of the 2 triangles meet at their homothety center. But the lines A'A, B'B, C'C are the bisectors of the angles $\angle A', \angle B', \angle C'$ (and also of the angles $\angle A, \angle B, \angle C$), hence, the homothety center is the common incenter I of the triangles $\triangle A'B'C',\ \triangle ABC$. The circumcenters O', O are the corresponding points of these 2 centrally similar triangles, hence, the line O'O also passes through the homothety center I.

Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$