Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, April 2014

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## Problems with sign 'C'

Deadline expired on May 12, 2014.

C. 1224. The integer that consists of $\displaystyle m$ digits of 9 and the integer that consists of $\displaystyle n$ digits of 9 are multiplied together. What is the sum of the digits of the product?

(5 pont)

solution (in Hungarian), statistics

C. 1225. What is the perimeter of the isosceles triangle that has a base of 6 cm and an inradius of 1.5 cm?

(5 pont)

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C. 1226. Solve the following equation on the set of pairs of integers: $\displaystyle x^2-3y^2+2xy-2x-10y+20=0$.

(5 pont)

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C. 1227. The lengths of the bases of a trapezium are 7 and 1. The trapezium is divided into two parts of equal area with a line parallel to the bases. Find the length of the segment of this parallel line that lies inside the trapezium.

(5 pont)

solution (in Hungarian), statistics

C. 1228. All cups are different in our kitchen cupboard. One third of them has the handle broken off. How many cups have we got if the number of ways to select two cups without handles and three with handles is exactly 1200?

(5 pont)

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C. 1229. A sphere is cut with the plane whose distance from the centre is 2/3 of the radius. What fraction of the volume is cut off?

(5 pont)

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C. 1230. Three lattice points are selected at random out of the lattice points lying on the circle of equation $\displaystyle x^2+y^2-2x-4y-45=0$. What is the probability that the three points form a right-angled triangle?

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on May 12, 2014.

B. 4622. The numbers $\displaystyle 1,2,\ldots,9$ are written in the fields of a $\displaystyle 3\times 3$ table so that the sum of the numbers is the same in each of the four $\displaystyle 2\times 2$ squares. What may this sum be?

(5 pont)

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B. 4623. In a convex quadrilateral, the diagonals form four triangles of integer areas. Prove that the product of the four integers cannot end in 2014.

(3 pont)

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B. 4624. In a trapezium $\displaystyle ABCD$, let $\displaystyle E$ and $\displaystyle F$ denote the midpoints of the bases $\displaystyle AB$ and $\displaystyle CD$, respectively, and let $\displaystyle O$ denote the intersection of the diagonals. A line parallel to the bases intersects the line segments $\displaystyle OA$, $\displaystyle OE$ and $\displaystyle OB$ at points $\displaystyle M$, $\displaystyle N$ and $\displaystyle P$, respectively. Show that the quadrilaterals $\displaystyle APCN$ and $\displaystyle BNDM$ have equal areas.

Suggested by L. Longáver, Nagybánya

(3 pont)

solution (in Hungarian), statistics

B. 4625. How many ordered pairs $\displaystyle (A,B)$ are there such that $\displaystyle A$ and $\displaystyle B$ are subsets of a fixed $\displaystyle n$-element set, and $\displaystyle A\subseteq B$?

(4 pont)

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B. 4626. Prove that $\displaystyle {(1+a)}^4 {(1+b)}^4 \ge 64ab {(a+b)}^2$ for all $\displaystyle a,b\ge 0$.

(6 pont)

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B. 4627. The angle bisector drawn from right-angled vertex $\displaystyle C$ of triangle $\displaystyle ABC$ intersects the circumscribed circle at point $\displaystyle P$, and the angle bisector from vertex $\displaystyle A$ intersects the circumscribed circle at point $\displaystyle Q$. $\displaystyle K$ is the intersection of line segments $\displaystyle PQ$ and $\displaystyle AB$. The centre of the inscribed circle is $\displaystyle O$, and its point of tangency on side $\displaystyle AC$ is $\displaystyle E$. Prove that points $\displaystyle E$, $\displaystyle O$ and $\displaystyle K$ are collinear.

Suggested by Zs. Sárosdi, Veresegyház

(4 pont)

solution (in Hungarian), statistics

B. 4628. Show that if $\displaystyle \alpha$, $\displaystyle \beta$ and $\displaystyle \gamma$ are the angles of a triangle then $\displaystyle \sin\alpha\cdot \sin\beta\cdot \cos\gamma+\sin\alpha\cdot \cos\beta\cdot \sin\gamma+ \cos\alpha\cdot \sin\beta\cdot \sin\gamma\le \frac 98$.

(4 pont)

solution (in Hungarian), statistics

B. 4629. Solve the equation $\displaystyle 2\sin\frac{3x}{2}=3\sin \left(x+\frac{\pi}{3}\right)$.

(5 pont)

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B. 4630. Points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$ are not coplanar. Determine the locus of the points $\displaystyle P$ for which $\displaystyle PA^{2}+PC^{2}=PB^{2}+PD^{2}$.

(5 pont)

solution (in Hungarian), statistics

B. 4631. The circles $\displaystyle k_0$, $\displaystyle k_1$, $\displaystyle k_2$, $\displaystyle k_3$ lie in the same plane and pairwise touch each other on the outside. The point of contact of $\displaystyle k_i$ and $\displaystyle k_j$ is $\displaystyle T_{ij}$. Let $\displaystyle k_0$ have centre $\displaystyle O$ and radius $\displaystyle r$, and let the circle $\displaystyle T_{12}T_{23}T_{31}$ have centre $\displaystyle U$ and radius $\displaystyle R$. Prove that $\displaystyle OU^2 = R^2-4Rr+r^2$.

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on May 12, 2014.

A. 614. In the triangle $\displaystyle A_1A_2A_3$, denote by $\displaystyle k_i$ the excircle opposite to $\displaystyle A_i$, and let $\displaystyle P_i$ be the point of $\displaystyle k_i$ for which the circle $\displaystyle A_{i+1}A_{i+2}P_i$ is tangent to $\displaystyle k_i$. ($\displaystyle i=1,2,3$; the indices are considered modulo $\displaystyle 3$.) Show that the line segments $\displaystyle A_1P_1$, $\displaystyle A_2P_2$ and $\displaystyle A_3P_3$ are concurrent.

(5 pont)

statistics

A. 615. Basil and Peter play the following game. Basil writes 100 real numbers on the board. After that they move alternately; Peter is first. In every move, the next player chooses two numbers, erases them and replaces both of them by their mean. Peter wins if he can achieve that the sum of suitably chosen 50 numbers is equal to the sum of the other 50 numbers. Determine whether Basil can prevent this.

Proposed by: I. Bogdanov and A. Shapovalov

(5 pont)

statistics

A. 616. Prove that

$\displaystyle \left(\frac{1+a}2\right)^{2x(x+y)} \left(\frac{1+b}2\right)^{2y(x+y)} \ge a^{x^2} b^{y^2} \left(\frac{a+b}2\right)^{2xy}$

holds for all real numbers $\displaystyle a,b>0$ and $\displaystyle x$, $\displaystyle y$.

(5 pont)

statistics