ISL 1966

Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?

Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove $$\left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.$$

A regular triangular prism has the altitude $h,$ and the two bases of the prism are equilateral triangles with side length $a.$ Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. Assume that a ray of light, entering the prism through the dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the prism through the dream-hole in the lower base. Find the angle between the upper base and the light ray at the moment when the light ray entered the prism, and the length of the way of the light ray in the interior of the prism.

Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right)$ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right)$ is the volume of the solid $M\left( R\right) .$ a.) Prove that $$V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.$$

Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.)

additional question:

b.) Find the area of the planar $R$ -neighborhood of a convex or non-convex polygon $m.$ c.) Find the volume of the $R$ -neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.Note by Darij: I guess that the '' $R$ -neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ?

Posted already on the board I think...

Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$

Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.

a.) Prove that the quadrilateral $MNPQ$ is a parallelogram.

b.) What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?

(Consecutive vertices of the parallelograms are labelled in alphabetical order.

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p},$ where $p$ is a real parameter. Investigate for which values of $p$ solutions exist and how many solutions exist.

(Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo $2\pi$ ''.)

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.

a.) What is the volume of this polyhedron ?

b.) Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$

Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $P^{\prime }.$

There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)

Prove that $$\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.$$

Prove the inequalitya.) $\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,$ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers.b.) Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality

$$a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },$$

then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of

a.) all vertices $A$ of such triangles;

b.) all vertices $B$ of such triangles;

c.) all vertices $C$ of such triangles.

A given natural number $N$ is being decomposed in a sum of some consecutive integers.

a.) Find all such decompositions for $N=500.$

b.) How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta$ and $\gamma$ are natural numbers) have? Which of these decompositions contain natural summands only?

c.) Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$

Note by Darij: The $0$ is not considered as a natural number.

The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$ ). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$ ). [Hereby, $a=BC,$ $b=CA,$ $c=AB,$ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$ ] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$ Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle. From each of the vertices of this triangle, we draw a tangent to the smaller circle. Prove that the length of one of these tangents equals the sum of the lengths of the two other tangents.

Find all pairs of positive integers $\left( x; y\right)$ satisfying the equation $2^{x}=3^{y}+5.$

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.

Note by Darij: A cyclic quadrilateralis a quadrilateral inscribed in a circle.

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle.

a.) Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle.

b.) Show that generally, one gets two possible points $P$ ( $P_{1}$ and $P_{2}$ ) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

Given a regular $n$ -gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$ ) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$

Note by Darij: Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color.

a.) Show that:

(1) Among the four segments originating at any of the $5$ points, two are red and two are blue.

(2) The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.)

b.) Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.

An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality $S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.)

Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.