Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up?
New to KöMaL?

KöMaL Problems in Mathematics, March 2021

Please read the rules of the competition.


Show/hide problems of signs:


Problems with sign 'K'

Deadline expired on April 12, 2021.


K. 689. In the 6th, 7th, 8th and 9th games of the season, a basketball player scored 23, 14, 11 and 20 points, respectively. His points average was higher after the 9th game than after the 5th game. With the 10th game, his average rose above 18. What is the lowest possible number of points that he may have scored in the 10th game?

(6 pont)

solution (in Hungarian), statistics


K. 690. Having a positive integer in mind, Peti formulated twenty-three statements about it. Two consecutive statements are false, but the rest of them are true:
1. It is divisible by 2.
2. It is divisible by 3.
3. It is divisible by 4.
\(\displaystyle \vdots\)
23. It is divisible by 24.

Peti was thinking about the smallest such number. What is his number?

(6 pont)

solution (in Hungarian), statistics


K. 691. \(\displaystyle ABCDEFGH\) is a regular octagon and its sides are 2 units long. Squares \(\displaystyle BCIM\) and \(\displaystyle FGKL\) are drawn on sides \(\displaystyle BC\) and \(\displaystyle GF\), on the inside. What is the area of the rectangle bounded by lines \(\displaystyle AH\), \(\displaystyle KL\), \(\displaystyle ED\) and \(\displaystyle IM\)?

(6 pont)

solution (in Hungarian), statistics


K. 692. A \(\displaystyle 6\times6\) square is dissected into lattice rectangles. What is the largest possible number of noncongruent rectangles obtained? Give an example.

(6 pont)

solution (in Hungarian), statistics


K. 693. Quadrilateral \(\displaystyle ABCD\) has an inscribed circle centred at \(\displaystyle O\). Show that the sum of \(\displaystyle \angle DOC\) and \(\displaystyle \angle BOA\) is \(\displaystyle 180^{\circ}\).

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'C'

Deadline expired on April 12, 2021.


C. 1658. A circular disc is divided into six congruent sectors. A circle is inscribed in each sector. The circle touches the arc of the sector as well as the two radii. What fraction of the area of the large circle is covered by the six smaller circles?

(5 pont)

solution (in Hungarian), statistics


C. 1659. Ray \(\displaystyle a\) starts from point \(\displaystyle A\) of a line segment \(\displaystyle AB\), and encloses an angle \(\displaystyle 0^\circ<\alpha<90^{\circ}\) with it. Ray \(\displaystyle b\) starts from point \(\displaystyle B\), and encloses an angle \(\displaystyle 0^\circ<\beta<90^{\circ}\) with the line segment \(\displaystyle AB\). The two rays lie in two different half planes of a plane containing line \(\displaystyle AB\). The circle of diameter \(\displaystyle AB\) intersects \(\displaystyle a\) again at \(\displaystyle A_1\), and \(\displaystyle b\) at \(\displaystyle B_1\). The circle of diameter \(\displaystyle A_1B_1\) intersects the line containing \(\displaystyle a\) again at \(\displaystyle A_2\), and the line containing \(\displaystyle b\) at \(\displaystyle B_2\). What is the relationship between \(\displaystyle \alpha\) and \(\displaystyle \beta\) if \(\displaystyle A_1B_1\) and \(\displaystyle A_2B_2\) are perpendicular?

(5 pont)

solution (in Hungarian), statistics


C. 1660. The positive integers \(\displaystyle 1\) to \(\displaystyle 61^2\) are written in the fields of a \(\displaystyle 61\times61\) chessboard, starting from the top left corner and proceeding along each row in succession. Then some changes are made as follows. In the first move, the sign of each number is changed to negative. In the second move, the signs of all even numbers are changed. In the third move, the sign of every multiple of 3 is changed, and so on, while the moves are meaningful. When all this is completed, how many \(\displaystyle 1\times2\) rectangles will there be on the chessboard in which the sum of the numbers is negative?

(5 pont)

solution (in Hungarian), statistics


C. 1661. In a lottery game, the player bets 5 numbers out of the positive integers 1 to 90. Otto Lotter insists on increase, and he always keeps the following rules when making his bets: he marks 5 numbers such that every digit may only occur once, and if the five numbers are listed in increasing order, the digits must be ascending, too. For example, 1, 2, 3, 46, 78. How many suitable selections of five numbers are there?

Proposed by Berkó Erzsébet, Szolnok

(5 pont)

solution (in Hungarian), statistics


C. 1662. For what values of the real parameter \(\displaystyle a>0\) will the equation \(\displaystyle x^2+a=\sqrt{x-a}\) have exactly one solution in the set of real numbers? What is the solution of the equation in that case?

(5 pont)

solution (in Hungarian), statistics


C. 1663. The circles \(\displaystyle k_1\) and \(\displaystyle k_2\) touch each other externally at point \(\displaystyle E\). Lines \(\displaystyle f\) and \(\displaystyle g\) pass through point \(\displaystyle E\). One of the common external tangents touches the circle \(\displaystyle k_1\) and \(\displaystyle k_2\) at points \(\displaystyle C\) and \(\displaystyle D\), respectively. Line \(\displaystyle h\) is obtained by dropping perpendiculars from point \(\displaystyle C\) onto lines \(\displaystyle f\) and \(\displaystyle g\), and connecting the feet of the perpendiculars. Line \(\displaystyle m\) is obtained in the same way, with perpendiculars from point \(\displaystyle D\). Prove that \(\displaystyle h\) and \(\displaystyle m\) are perpendicular to each other.

(5 pont)

solution (in Hungarian), statistics


C. 1664. Each of the diagonals \(\displaystyle AD\), \(\displaystyle BE\) and \(\displaystyle CF\) of a convex hexagon \(\displaystyle ABCDEF\) halves the area of the hexagon. Prove that these diagonals are concurrent.

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on April 12, 2021.


B. 5158. Let \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\) and \(\displaystyle D\) be points in the plane such that \(\displaystyle AB<CB\) and \(\displaystyle CD<AD\). Prove that the line segments \(\displaystyle AB\) and \(\displaystyle CD\) do not intersect each other.

(3 pont)

solution (in Hungarian), statistics


B. 5159. Solve the equation

\(\displaystyle \left[\frac{2020-x}{x-1}\right]+\left[\frac{2021+x}{x+1}\right]=82 \)

over the set of integers, where \(\displaystyle [c]\) denotes the greatest integer not greater than \(\displaystyle c\).

Proposed by Zs. M. Tatár, Esztergom

(4 pont)

solution (in Hungarian), statistics


B. 5160. What may be the value of \(\displaystyle x+y+z\) if

\(\displaystyle \sqrt{x-1}+2\sqrt{y-4}+3\sqrt{z-9}=\frac{x+y+z}{2}? \)

Proposed by M. Szalai, Szeged

(3 pont)

solution (in Hungarian), statistics


B. 5161. \(\displaystyle 800\) L-tetrominoes are laid on a \(\displaystyle 100\times 100\) chesssboard. Prove that it is possible to place another L-tetromino on the board. The tetrominoes do not overlap, and each of them covers exactly \(\displaystyle 4\) fields of the chessboard. An L-tetromino is defined as the shape shown in the figure, including any rotated or reflected images.

(6 pont)

solution (in Hungarian), statistics


B. 5162. The sides of triangle \(\displaystyle ABC\) are 9, 10 and 17 units long. What is the area of the triangle formed by the exterior angle bisectors of triangle \(\displaystyle ABC\)?

Proposed by Zs. M. Tatár, Esztergom

(5 pont)

solution (in Hungarian), statistics


B. 5163. Triangle \(\displaystyle ABC\) is right-angled at \(\displaystyle C\). The right angle is divided \(\displaystyle 2:1\) by a line such that the smaller part is closer to the shorter leg. This line intersects hypotenuse \(\displaystyle AB\) at \(\displaystyle T\), and the circumscribed circle at \(\displaystyle D\). What are the measures of the acute angles of the triangle if the feet of the perpendiculars dropped from \(\displaystyle D\) onto the lines of the legs are collinear with \(\displaystyle T\)?

(4 pont)

solution (in Hungarian), statistics


B. 5164. Two players continue playing rock-paper-scissors games until one of them wins the third time. Assume that each player in each game selects rock, paper or scissors at random (independently of each other and of previous games), with probabilities of \(\displaystyle \frac13 : \frac13 : \frac13\). Determine the expected value of the number of games needed.

(5 pont)

solution (in Hungarian), statistics


B. 5165. Given a positive integer \(\displaystyle k\), is there a function \(\displaystyle f \colon \mathbb{N} \to \mathbb{N}\), such that

\(\displaystyle f(x) + f\big(f(x)\big) = x + k \)

for all \(\displaystyle x \in \mathbb{N}\)?

Proposed by M. Lovas, Budapest

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on April 12, 2021.


A. 795. The following game is played with a group of \(\displaystyle n\) people: \(\displaystyle n+1\) hats are numbered from \(\displaystyle 1\) to \(\displaystyle n+1\). The people are blindfolded, and each of them is getting one of the \(\displaystyle n+1\) hats on his head (the remaining hat is hidden). Now a line is formed from the \(\displaystyle n\) people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players).

What is the highest number of guaranteed correct guesses, if the \(\displaystyle n\) people can discuss a common strategy after learning about the game?

Submitted by Viktor Kiss, Budapest

(7 pont)

statistics


A. 796. Let \(\displaystyle ABCD\) be a cyclic quadrilateral. Let lines \(\displaystyle AB\) and \(\displaystyle CD\) intersect in \(\displaystyle P\), and lines \(\displaystyle BC\) and \(\displaystyle DA\) intersect in \(\displaystyle Q\). The feet of the perpendiculars from \(\displaystyle P\) to \(\displaystyle BC\) and \(\displaystyle DA\) are \(\displaystyle K\) and \(\displaystyle L\), and the feet of the perpendiculars from \(\displaystyle Q\) to \(\displaystyle AB\) and \(\displaystyle CD\) are \(\displaystyle M\) and \(\displaystyle N\). The midpoint of diagonal \(\displaystyle AC\) is \(\displaystyle F\).

Prove that the circumcircles of triangles \(\displaystyle FKN\) and \(\displaystyle FLM\), and the line \(\displaystyle PQ\) are concurrent.

Based on a problem by Ádám Péter Balogh, Szeged

(7 pont)

statistics


Upload your solutions above.