MathNet 1968 — Part 01

The medians divide a triangle into 6 smaller triangles. 4 of the circles inscribed in the smaller triangles have equal radii. Prove that the original triangle is equilateral.

$\{a_i\}$ and $\{b_i\}$ are permutations of $\{1/1, 1/2, \dots, 1/n\}$. $a_1 + b_1 \geq a_2 + b_2 \geq \ldots \geq a_n + b_n$. Prove that for every $m$ ($1 \leq m \leq n$), $a_m + b_m \geq \dfrac{4}{m}$.

An octagon has equal angles. The lengths of the sides are all integers. Prove that the opposite sides are equal in pairs.

The sequence $\{a_n\}$ is defined as follows: $a_1 = 1$, $a_{n+1} = a_n + 1 / a_n$ for $n \geq 1$. Prove that $a_{100} \gt 14$.

Given a triangle $ABC$, and $D$ on the segment $AB$, $E$ on the segment $AC$, such that $AD = DE = AC$, $BD = AE$, and $DE$ is parallel to $BC$. Prove that $BD$ equals the side of a regular 10-gon inscribed in a circle with radius $AC$.

The difference between the longest and shortest diagonals of the regular $n$-gon equals its side. Find all possible $n$.

Prove that we can find positive integers $x$, $y$ satisfying $x^{2} + x + 1 = p y$ for an infinite number of primes $p$.

Which is greater: $31^{11}$ or $17^{14}$? [No calculators allowed!]

Given a regular tetrahedron $ABCD$, prove that it is contained in the three spheres on diameters $AB$, $BC$ and $AD$. Is this true for any tetrahedron?

A circle radius $100$ is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares can it pass through?

The circle inscribed in the triangle $ABC$ touches the side $AC$ at $K$. Prove that the line joining the midpoint of $AC$ with the center of the circle bisects the segment $BK$.

There is a set of scales on the table and a collection of weights. Each weight is on one of the two pans. Each weight has the name of one or more pupils written on it. All the pupils are outside the room. If a pupil enters the room then he moves the weights with his name on them to the other pan. Show that you can let in a subset of pupils one at a time, so that the scales change position after the last pupil has moved his weights.

In a group of students, $50$ speak English, $50$ speak French and $50$ speak Spanish. Some students speak more than one language. Prove it is possible to divide the students into $5$ groups (not necessarily equal), so that in each group $10$ speak English, $10$ speak French and $10$ speak Spanish.

The sides and diagonals of $ABCD$ have rational lengths. The diagonals meet at $O$. Prove that the length $AO$ is also rational.

Prove that:
$$
\frac{2}{x^{2} - 1} + \frac{4}{x^{2} - 4} + \frac{6}{x^{2} - 9} + \ldots + \frac{20}{x^{2} - 100} =
$$
$$
\frac{11}{(x - 1)(x + 10)} + \frac{11}{(x - 2)(x + 9)} + \ldots + \frac{11}{(x - 10)(x + 1)}.
$$

Given point $O$ inside the acute-angled triangle $ABC$, and point $O'$ inside the acute-angled triangle $A'B'C'$. $D$, $E$, $F$ are the feet of the perpendiculars from $O$ to $BC$, $CA$, $AB$ respectively, and $D'$, $E'$, $F'$ are the feet of the perpendiculars from $O'$ to $B'C'$, $C'A'$, $A'B'$ respectively. $OD$ is parallel to $O'A'$, $OE$ is parallel to $O'B'$ and $OF$ is parallel to $O'C'$. Also $OD - O'A' = OE - O'B' = OF - O'C'$. Prove that $O'D'$ is parallel to $OA$, $O'E'$ to $OB$ and $O'F'$ to $OC$, and that $O'D' \cdot OA = O'E' \cdot OB = O'F' \cdot OC$.

The streets in a city are on a rectangular grid with $m$ east-west streets and $n$ north-south streets. It is known that a car will leave some (unknown) junction and move along the streets at an unknown and possibly variable speed, eventually returning to its starting point without ever moving along the same block twice. Detectors can be positioned anywhere except at a junction to record the time at which the car passes and its direction of travel. What is the minimum number of detectors needed to ensure that the car's route can be reconstructed?

9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the winner could have obtained?