Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, April 2017

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## Problems with sign 'C'

Deadline expired on May 10, 2017.

C. 1413. Find all natural numbers $\displaystyle n$ less than 300 for which $\displaystyle \frac{n(n+1)}{2}$ is a square number.

(5 pont)

solution (in Hungarian), statistics

C. 1414. Let $\displaystyle E$ and $\displaystyle F$ denote the points that divide side $\displaystyle CD$ of a unit square $\displaystyle ABCD$ into three equal parts. Lines $\displaystyle BE$ and $\displaystyle BF$ intersect diagonal $\displaystyle AC$ at $\displaystyle K$ and $\displaystyle L$. Find the exact length of $\displaystyle KL$.

(Matlap, Kolozsvár)

(5 pont)

solution (in Hungarian), statistics

C. 1415. A number is written on a blackboard. Two players take turns in selecting one digit of the number on the board, subtracting it from the number, erasing the number on the board and replacing it with the difference obtained. The winner is the player who finally writes 0 on the board. Which player has a winning strategy, and what is the winning strategy if the number they start with is 2017?

(Matlap, Kolozsvár)

(5 pont)

solution (in Hungarian), statistics

C. 1416. How many position vectors of length 10 are there for which three times the $\displaystyle x$ coordinate is greater than four times the $\displaystyle y$ coordinate, and at least one coordinate is an integer?

(5 pont)

solution (in Hungarian), statistics

C. 1417. Solve the simultaneous equations

$\displaystyle a + b = c + d,$

$\displaystyle \frac 1a + \frac 1b = \frac 1c + \frac 1d.$

(Proposed by Á. Kertész)

(5 pont)

solution (in Hungarian), statistics

C. 1418. Six consecutive integers are found to have the following interesting property: the sum of the six numbers is a prime, and the sum of their squares is also a prime. Find all sets of six numbers with this property.

(5 pont)

solution (in Hungarian), statistics

C. 1419. In an acute-angled triangle $\displaystyle ABC$, consider the two circular sectors, each bounded by an arc $\displaystyle AB$, with one arc passing through the foot of the altitude drawn from $\displaystyle A$, and the other arc passing through the orthocentre of the triangle. Prove that if $\displaystyle \angle ACB =45^\circ$ then the areas of the two circular sectors are equal.

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on May 10, 2017.

B. 4867. The sum of the real numbers $\displaystyle a$, $\displaystyle b$, $\displaystyle c$ and $\displaystyle d$ is $\displaystyle 0$. Let $\displaystyle M=ab+bc+cd$ and $\displaystyle N=ac+ad+bd$. Prove that at least one of the sums $\displaystyle 20M+17N$ and $\displaystyle 20N+17M$ is non-positive.

(Bulgarian problem)

(4 pont)

solution (in Hungarian), statistics

B. 4868. In a triangle $\displaystyle ABC$, $\displaystyle AC<AB$ and the median $\displaystyle AF$ divides the angle at $\displaystyle A$ in a $\displaystyle 1:2$ ratio. The perpendicular drawn to $\displaystyle AB$ at $\displaystyle B$ intersects line $\displaystyle AF$ at $\displaystyle D$. Show that $\displaystyle AD=2AC$.

(3 pont)

solution (in Hungarian), statistics

B. 4869. Let $\displaystyle A$ be a finite set of real numbers. A partition of the elements of $\displaystyle A$ into at least two subsets is said to be a tiling if the (pairwise disjoint) subsets have at least two elements each, and are obtained from each other by a translation. Prove that $\displaystyle A$ has an even number of tilings.

(5 pont)

solution (in Hungarian), statistics

B. 4870. The diagonals of a convex quadrilateral $\displaystyle ABCD$ intersect at point $\displaystyle E$. Prove that the centres of the Feuerbach circles of the triangles $\displaystyle ABE$, $\displaystyle BCE$, $\displaystyle CDE$ and $\displaystyle DAE$ either form a parallelogram or are collinear.

(Proposed by Sz. Miklós, Herceghalom)

(5 pont)

solution (in Hungarian), statistics

B. 4871. Prove that the number $\displaystyle a_n=1001001\ldots 1001$ (where $\displaystyle n$ denotes the number of ones) cannot be a prime.

(3 pont)

solution (in Hungarian), statistics

B. 4872. Let $\displaystyle p$, $\displaystyle q$ and $\displaystyle r$ denote three distinct primes, and let $\displaystyle n=pq^2r^3$. Show that

$\displaystyle (n,1)+(n,2)+\dots+(n,n) =qr^2(2p-1)(3q-2)(4r-3),$

where $\displaystyle (a,b)$ denotes the greatest common divisor of $\displaystyle a$ and $\displaystyle b$.

(Proposed by J. Szoldatics, Budapest)

(6 pont)

solution (in Hungarian), statistics

B. 4873. In a triangle $\displaystyle ABC$, $\displaystyle AB=1$, $\displaystyle \angle BAC =135^{\circ}$, $\displaystyle \angle ABC =30^{\circ}$. Find the parameter of the parabola with symmetry axis $\displaystyle AB$ that is tangent to the lines $\displaystyle AC$ and $\displaystyle BC$.

(5 pont)

solution (in Hungarian), statistics

B. 4874. Let $\displaystyle \|x\|$ denote the distance of the real number $\displaystyle x$ from the closest integer. Let $\displaystyle a_1,a_2,\ldots,a_n$ be positive integers. Prove that there exists a real number $\displaystyle x$ such that $\displaystyle \|a_ix\|\ge \frac{1}{2n}$ for all $\displaystyle 1\le i\le n$.

(5 pont)

solution (in Hungarian), statistics

B. 4875. Line $\displaystyle g$ encloses an angle $\displaystyle \alpha$ with the plane determined by lines $\displaystyle e$ and $\displaystyle f$. Show that

$\displaystyle \angle(e,f)+\angle(e,g)+\angle(g,f)\le \pi +\alpha.$

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on May 10, 2017.

A. 695. We are given $\displaystyle 2k$ lines, $\displaystyle e_1, \ldots, e_{2k}$ in a plane $\displaystyle S$, further given a line $\displaystyle g$ which has an angle $\displaystyle \alpha$ with $\displaystyle S$. Show that the sum of the pairwise angles between the lines $\displaystyle e_1, \ldots,e_{2k}, g$ is at most

$\displaystyle (k^2+k)\cdot \dfrac \pi 2 + k\alpha.$

(5 pont)

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A. 696. Let $\displaystyle k\ge2$ be an integer. Determine all those polynomials $\displaystyle p(x)$ with real coefficients for which

$\displaystyle p(x) \cdot p(2x^k-1) = p(x^k) \cdot p(2x-1).$

(5 pont)

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A. 697. For all primes $\displaystyle p\ge3$, let

$\displaystyle S(p) = \sum_{k=1}^{\frac{p-1}2} \tan \frac{k^2\pi}{p}.$

$\displaystyle (a)$ Show that $\displaystyle p\equiv1\pmod4$ implies $\displaystyle S(p)=0$.

$\displaystyle (b)$ Show that if $\displaystyle p\equiv3\pmod4$, then $\displaystyle \dfrac{S(p)}{\sqrt{p}}$ is an odd integer.

(5 pont)

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