Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, May 2016

Please read the rules of the competition.

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## Problems with sign 'C'

Deadline expired on June 10, 2016.

C. 1357. Sophie had a rectangular prism with a square base and one edge is 3 cm longer than the other edge. She told Greg the surface area of the prism. Given that the lengths of the edges in centimetres are integers, Greg was able to find out the lengths of the edges. How long are the edges?

(5 pont)

solution, statistics

C. 1358. The pentagon $\displaystyle ABCDE$ has an inscribed circle of centre $\displaystyle O$ and radius $\displaystyle r$. Given that the angle at vertex $\displaystyle A$ is a right angle, $\displaystyle \angle EOA=60^\circ$, and that triangle $\displaystyle OCD$ is equilateral, find the area of the pentagon.

(5 pont)

solution, statistics

C. 1359. Depending on the positive integer $\displaystyle n$, how many solutions does the equation

$\displaystyle x^2-y^2=10^n$

have on the set of pairs of non-negative integers $\displaystyle (x;y)$?

(5 pont)

solution, statistics

C. 1360. How many sides does a regular polygon have if it has a diagonal whose Thales circle passes through the midpoint of a side?

(5 pont)

solution, statistics

C. 1361. In the equation $\displaystyle ax^2+bx+c=0$, the positive integers $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, in this order, form an arithmetic progression. The roots of the equation are also integers. What are the values of the two roots?

Proposed by Á. Kertész

(5 pont)

solution, statistics

C. 1362. The volume of a cuboid is 10.9545 cm$\displaystyle {}^3$, the arithmetic mean of the lengths of the edges is 2.2655 cm, and their harmonic mean is 2.1769 cm. Determine the length of the diagonal of the cuboid, to the nearest thousandth of a centimetre.

(5 pont)

solution, statistics

C. 1363. The parallel sides of a right-angled trapezium are $\displaystyle 405$ and $\displaystyle 80$ units long. The length of the right-angled leg is $\displaystyle 65 \big(\sqrt{3}+\sqrt{2}\,\big)$ units. The trapezium is sliced into eight trapezoids, all similar to one another, with cuts parallel to the bases. What is the height of the largest slice?

Proposed by Z. G. Szepesi, Budapest

(5 pont)

solution, statistics

## Problems with sign 'B'

Deadline expired on June 10, 2016.

B. 4795. The hour hand and the minute hand of a clock are identical, and it has no second hand. How many time instants are there after noon and before midnight when it is impossible to tell the time by looking at the clock?

Proposed by M. E. Gáspár, Budapest

(3 pont)

solution, statistics

B. 4796. Solve the following equation on the set of real numbers:

$\displaystyle x^2-6\{x\}+1=0,$

where $\displaystyle \{x\}$ stands for the fractional part of a number $\displaystyle x$ (that is, the difference obtained when the largest integer not greater than $\displaystyle x$ is subtracted from $\displaystyle x$).

Proposed by J. Szoldatics, Budapest

(4 pont)

solution, statistics

B. 4797. In triangle $\displaystyle ABC$, $\displaystyle D$, $\displaystyle E$ and $\displaystyle F$ are arbitrary interior points of sides $\displaystyle AB, BC$ and $\displaystyle CA$, respectively. Let $\displaystyle G$, $\displaystyle H$ and $\displaystyle I$ denote the centroids of triangles $\displaystyle ADF$, $\displaystyle BED$ and $\displaystyle CFE$, respectively. Furthermore, let $\displaystyle S$, $\displaystyle K$, $\displaystyle L$ be the centroids of triangles $\displaystyle ABC$, $\displaystyle DEF$ and $\displaystyle GHI$, respectively. Prove that the points $\displaystyle K$, $\displaystyle L$ and $\displaystyle S$ are collinear.

Proposed by Sz. Miklós, Herceghalom

(3 pont)

solution, statistics

B. 4798. In a cyclic quadrilateral $\displaystyle ABCD$, diagonals $\displaystyle AC$ and $\displaystyle BD$ are perpendicular, and the centre of the circumscribed circle is $\displaystyle K$. Prove that the areas of triangles $\displaystyle ABK$ and $\displaystyle CDK$ are equal.

(4 pont)

solution, statistics

B. 4799. Determine those pairs of non-negative integers $\displaystyle (a,b)$ for which $\displaystyle 5^a+3^b$ is a perfect square.

Proposed by T. Káspári, Paks

(5 pont)

solution, statistics

B. 4800. $\displaystyle T$ is a point on line $\displaystyle BC$, different from the midpoint of line segment $\displaystyle BC$. Circle $\displaystyle k$ is centred at $\displaystyle T$, and $\displaystyle A$ is its intersection with the perpendicular drawn to $\displaystyle BC$ at $\displaystyle T$. The intersections of $\displaystyle k$ with the lines $\displaystyle AB$ and $\displaystyle AC$ are $\displaystyle K$ and $\displaystyle L$, respectively. Let $\displaystyle k$ intersect the circumscribed circle of $\displaystyle ABC$ again at $\displaystyle M$. Prove that the lines $\displaystyle KL$, $\displaystyle AM$ and $\displaystyle BC$ are concurrent.

Proposed by K. Williams, Szeged

(5 pont)

solution, statistics

B. 4801. Define the sequence $\displaystyle f_n$ of functions by the following recurrence relation:

$\displaystyle f_0(x) = f_1(x) = 1, \mathrm{~and ~for ~} n\ge 2 \quad f_n(x) = f_{n-1}(x) \cdot 2\cos(2x) - f_{n-2}(x).$

Determine the number of roots of $\displaystyle f_n(x)$ in the interval $\displaystyle [0,\pi]$.

Proposed by L. Bodnár, Budapest

(5 pont)

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B. 4802. The inscribed sphere of a right circular cone $\displaystyle \mathcal{K}$ is $\displaystyle \mathcal{G}$. The centres of the spheres $\displaystyle g_1, g_2, \dots, g_n$ of radius $\displaystyle r$ form a regular $\displaystyle n$-gon of side $\displaystyle 2r$. Furthermore, each sphere $\displaystyle g_i$ is tangent to both the lateral surface and the base of $\displaystyle \mathcal{K}$, and also tangent to $\displaystyle \mathcal{G}$. What may be the value of $\displaystyle n$?

Competition problem from the Soviet Union

(6 pont)

solution, statistics

B. 4803. Is it possible to specify closed intervals of rational endpoints on the number line such that every rational number is the endpoint of exactly one interval, and

$\displaystyle a)$ one of any pair of closed intervals contains the other;

$\displaystyle b)$ no pair of two intervals are disjoint, but no interval contains another?

Based on an idea of M. E. Gáspár

(6 pont)

solution, statistics

## Problems with sign 'A'

Deadline expired on June 10, 2016.

A. 671. Prove that

$\displaystyle 0< \sum_{i=0}^k {(-1)}^{i}\binom{n+1}{i}{(k+1-i)}^n < n!$

holds for every pair $\displaystyle 0<k<n$ of integers.

(5 pont)

solution, statistics

A. 672. Point $\displaystyle O$ is the apex of an oblique circular cone. Show that there are some points $\displaystyle F_1$ and $\displaystyle F_2$ in the interior of the base such that $\displaystyle \angle XOF_1 +\angle XOF_2$ is constant when $\displaystyle X$ runs along the perimeter of the base disk.

(5 pont)

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A. 673. We have colour pearls placed on an $\displaystyle n\times n$ board; a square may contain more than one pearl. Altogether we used $\displaystyle 2n-1$ colours and $\displaystyle n$ pearls from each colour. The pearls are arranged in such a way that no row or column contains more than one pearl of the same colour. Prove that it is possible to select $\displaystyle n$ pearls with distinct colours such that no two of them are in the same row or column.

(5 pont)

solution, statistics

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