BME math contest 2021

Problem nr. 1

Prove that the upper integer part of $(\sqrt{3}+1)^{2m}$ is divisible by $2^{m+1}$ for every $m>1$ integer! (8 points)

Problem nr. 2

Let $x_1, x_2, \ldots , x_n$ be positive numbers such that $x_1+x_2+\ldots + x_n \geq x_1^2 + x_2^2 + \ldots + x_n^2$. Show that $x_1^{x_1}x_2^{x_2}\ldots x_n^{x_n}\leq 1$ and moreover determine the case of equality! (9 points)

Problem nr. 3

A circular route has stations with fuel reserves. Prove that if the total amount of fuel (i.e. the combined reserves of the stations) is sufficient for our vehicle to complete the circle twice, then there exists a station from which we could make a round trip in either direction around the loop. (During the trip, we can only use fuel provided by the stations; we begin with an empty tank, and fuel usage is proportional to the distance travelled. The stations need not be equally spaced around the loop, and the amount of fuel waiting for us at each station can vary. Our vehicle has a tank large enough to accommodate the fuel needed for a full circle.) (11 points)

Problem nr. 4

On the surface of a 3-dimensional ball there are 2021 marked points, determining $2021 \choose 2$ line segments. (Here we really intend to mean line segments - passing through the interiour of the ball - and not arcs.) Among them, some could be shorter and some longer. Give an upper bound on the cardinality of the ones with length strictly greater than $\sqrt{\frac{8}{3}}$! (Finding the optimal bound - and proving its optimality - values 12 points. Giving suboptimal bounds or even the optimal one but without proving its optimality results in fewer points.)

Problem nr. 5

A natural number is colorful if all 10 possible values appear among its digits (written in base 10). Show that there exists a natural number $n$ such that the numbers $n, 2n, 3n, \ldots , 2021n$ are all colorful! Is there also an $n$ such that all (positive integer) multiples of $n$ are colorful? (8+4 points)

Problem nr. 6

Let $X,Y,Z$ be three matrices with sizes such that the matrix product $XYZ$ is well-defined and let $r(M)$ stand for the rank of a matrix $M$. Show that the inequality $$r(XY)+r(YZ)\leq r(XYZ)+r(Y)$$ holds! (11 points)

Problem nr. 7

Let $\alpha >0$ be a fixed parameter. Prove that any derivative of odd order of the function $f$ defined by the formula $f(x)=\frac{e^x}{x^\alpha}$ is monotonically increasing on the half-line $(0,\infty)$! (12 points)

Problem nr. 8

Consider an unbiased die with $n$ faces labeled from $1$ to $n$. We roll this die $n+1$ times and write down the results. Let $p_n$ be the probability that we can choose $n$ terms from the thus obtained list of $n+1$ numbers, whose sum is divisible by $n$. Consider the sequence $p_1,p_2,p_3,\ldots $ and determine its limit (if it exists at all)! (10 points)

Problem nr. 9

Let $n>1$ be an integer and $H$ the set of ordered pairs of positive numbers $(a,b)$ such that $a,b\leq n$, ${\rm gcd}(a,b)=1$ and $a+b>n$. Prove that the equality $$ \sum_{(a,b)\in H}\frac{1}{ab} = 1 $$ holds! (11 points)