Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, March 2015

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## Problems with sign 'K'

Deadline expired on April 10, 2015.

K. 457. (According to a Hungarian poem for children (http://www.mdche.u-szeged.hu/$\displaystyle \sim$kovacs/sheepshrine_f.html) it is a rule in the school for young sheep that students get an award for every day they do not attend the classes. On Monday, 10% of the students were absent. On Tuesday, 10% of those absent on Monday returned, but 10% of those present on Monday stayed at home. What percentage of the students will not get a reward for Tuesday?

(6 pont)

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K. 458. Johnny wanted to buy some nails. In one shop, where 100 grams of nails cost 180 forints (HUF, Hungarian currency), he could not buy the quantity needed since he was 1430 forints short. So he went to another shop where 100 grams only cost 120 forints. He bought the quantity he needed, and 490 forints still remained in his pocket. How many kilos of nails did he need?

(6 pont)

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K. 459. Uncle Charlie had a ladder of length 2 metres and 60 cm. In order to replace a light bulb in a lamp on the wall, he leaned the ladder against the wall, with the bottom end at a distance of 156 cm from the wall. It turned out that he would need to climb 32 cm higher in order to reach the bulb, so he pushed the bottom of the ladder closer to the wall. By how many centimetres did he push it closer?

(6 pont)

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K. 460. A circle of radius 10 units is centred at point $\displaystyle O$. $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are points on the circle such that $\displaystyle O$ lies in the interior of triangle $\displaystyle ABC$. Given that the length of line segment $\displaystyle AB$ is 12 units and the measure of angle $\displaystyle ABC$ is $\displaystyle 60^\circ$, find

$\displaystyle a)$ the distance of point $\displaystyle O$ from line segment $\displaystyle AB$,

$\displaystyle b)$ the length of line segment $\displaystyle AC$.

(6 pont)

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K. 461. A set of coordinate axes is drawn on a sheet of squared paper. Then the sheet is folded along a straight line such that the point $\displaystyle (30;12)$ is moved to the point $\displaystyle (-2;-4)$. Where does the folding line intersect the coordinate axes?

(6 pont)

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K. 462. $\displaystyle a)$ $\displaystyle f$ is a function defined on the set of real numbers. Given that $\displaystyle f(a)-f(b)=f(a\cdot b)$ for all $\displaystyle a$ and $\displaystyle b$, find the value of $\displaystyle f(2015)$.

$\displaystyle b)$ Is there a function $\displaystyle g$ defined on the set of real numbers such that $\displaystyle g(a) - g(b) = 2\cdot g(a\cdot b) - 2$ for all $\displaystyle a$ and $\displaystyle b$?

(6 pont)

solution, statistics

## Problems with sign 'C'

Deadline expired on April 10, 2015.

C. 1280. Given that the natural numbers $\displaystyle m$ and $\displaystyle n$ are relatively primes, prove that the greatest common factor of $\displaystyle m+n$ and $\displaystyle m^2+n^2$ is either 1 or 2.

(5 pont)

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C. 1281. Let $\displaystyle M$ denote the intersection of the lines of the legs of a trapezium. On a line passing through $\displaystyle M$ and parallel to the bases, let $\displaystyle A$ and $\displaystyle B$ denote the intersections with the extensions of the diagonals. Prove that $\displaystyle |AM|=|BM|$.

(5 pont)

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C. 1282. How many solutions does the equation $\displaystyle 2^{a}+3^{b}+4^{c}+5^{d}+6^{e}=22$ have where $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, $\displaystyle d$, $\displaystyle e$ are integers?

(5 pont)

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C. 1283. The longer base, $\displaystyle AB$, of trapezium $\displaystyle ABCD$ is not greater than three times the base $\displaystyle CD$. Lines $\displaystyle e$ and $\displaystyle f$ are parallel to the legs $\displaystyle BC$ and $\displaystyle DA$, respectively, and each of them halves the area of the trapezium. Let $\displaystyle P$ and $\displaystyle Q$ denote the intersections of $\displaystyle AB$ with $\displaystyle e$ and $\displaystyle f$, respectively, and let $\displaystyle P'$ and $\displaystyle Q'$ denote their intersections with $\displaystyle DC$.

$\displaystyle a)$ Prove that the intersection $\displaystyle M$ of lines $\displaystyle e$ and $\displaystyle f$ lies on the midline of the trapezium.

$\displaystyle b)$ Given that the quadrilateral $\displaystyle PQ'P'Q$ is a parallelogram, find the ratio of the area of triangle $\displaystyle MPQ$ to the area of the trapezium $\displaystyle ABCD$.

(5 pont)

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C. 1284. Five cards are drawn at random from a deck of German cards. Which of the following events has a higher probability: the event that all five are from the same suit, or the event that there are four cards among them with the same denomination?

(Competition problem from Germany)

(5 pont)

solution, statistics

C. 1285. The length of the radius of the inscribed circle of an isosceles triangle is divided by the length of the radius of the circumscribed circle. What is the maximum possible value of the ratio?

(5 pont)

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C. 1286. Solve the following simultaneous equations:

$\displaystyle y^2 =x^3-3x^2+2x,$

$\displaystyle x^2 =y^3-3y^2+2y.$

(5 pont)

solution, statistics

## Problems with sign 'B'

Deadline expired on April 10, 2015.

B. 4696. How many positive integers $\displaystyle n$ are there such that the geometric mean and the harmonic mean of $\displaystyle n$ and 2015 are both integers?

(3 pont)

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B. 4697. Prove that if a right-angled trapezium has an inscribed circle, then the shorter leg equals the line segment cut out by the legs from the line parallel to the bases, passing through the intersection of the diagonals.

(4 pont)

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B. 4698. Give an example for sets $\displaystyle H_1,H_2,\ldots\subset\mathbb{N}$ for which the following conditions hold:

$\displaystyle a)$ $\displaystyle |H_n|=n$ for all positive integers $\displaystyle n$.

$\displaystyle b)$ For all positive integers $\displaystyle n$ and $\displaystyle k$, $\displaystyle H_n \cap H_k = H_{(n,k)}$, where $\displaystyle (n,k)$ is the greatest common divisor of $\displaystyle n$ and $\displaystyle k$.

(5 pont)

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B. 4699. Construct a bicentric kite if given the radius of its circumscribed circle and the distance between the centres of its circumscribed and inscribed circles.

(4 pont)

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B. 4700. Solve the equation

$\displaystyle \big(\sqrt{1+\sin^2 x}-\sin x\big) \big(\sqrt{1+\cos^2 x}-\cos x\big) =1.$

(5 pont)

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B. 4701. Let $\displaystyle A_1 B_1 C_1 D_1$ be a quadrilateral. For any set of four points $\displaystyle A_n B_n C_n D_n$ already defined for a positive integer $\displaystyle n$, let $\displaystyle A_{n+1}$ be the centroid of the triangle $\displaystyle B_{n}C_{n}D_{n}$. $\displaystyle B_{n+1}$, $\displaystyle C_{n+1}$ and $\displaystyle D_{n+1}$ are defined analogously, with a cyclic permutation of the points. Show that for any starting quadrilateral, the point sequence $\displaystyle A_n$ only has a finite number of points lying outside the unit circle drawn about the centre of mass of the quadrilateral $\displaystyle A_1 B_1 C_1 D_1$.

Suggested by E. Gáspár Merse, Budapest

(4 pont)

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B. 4702. Show that there are infinitely many points equidistant from three pairwise skew edges of a given cube.

(5 pont)

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B. 4703. Given that the absolute values of the numbers $\displaystyle x_1$, $\displaystyle x_2$, $\displaystyle x_3$, $\displaystyle x_4$, $\displaystyle x_5$, $\displaystyle x_6$ are at most 1, and their sum is 0, prove that

$\displaystyle 3\sum_{i=1}^{5} {\sqrt{1-x_i^2}} \le \sum_{i=1}^{5} {\sqrt{9-{(x_i+x_{i+1})}^2}}\,.$

Suggested by K. Williams, Szeged

(6 pont)

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B. 4704. The circles $\displaystyle k_2$ and $\displaystyle k_3$ have different radii. A circle $\displaystyle k_1$ touches both of them from the inside. The circles $\displaystyle k_2$ and $\displaystyle k_3$ are tangent to a circle $\displaystyle k_4$ from the inside. Show that the radical axis of $\displaystyle k_1$ and $\displaystyle k_4$ passes through the external point of similitude of $\displaystyle k_2$ és $\displaystyle k_3$.

(6 pont)

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## Problems with sign 'A'

Deadline expired on April 10, 2015.

A. 638. Does there exist a simple, closed polyline in the 3-dimensional Cartesian co-ordinate system whose perpendicular projection onto every co-ordinate plane is a cycleless graph?

(5 pont)

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A. 639. The angle trisectors of a triangle bound a convex hexagon in the interior of the triangle. Prove that the diagonals of this hexagon, that connect opposite vertices, are concurrent.

Proposed by: Zoltán Bertalan, Békéscsaba

(5 pont)

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A. 640. Find all primes $\displaystyle p$ and positive integers $\displaystyle n$ for which the numbers $\displaystyle {(k+1)}^n-2k^n$ ($\displaystyle k=1,2,\ldots,p$) form a complete residue system modulo $\displaystyle p$.

(5 pont)

solution, statistics