Mathematical and Physical Journal
for High Schools
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KöMaL Problems in Mathematics, February 2021

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Problems with sign 'K'

Deadline expired on March 10, 2021.


K. 684. \(\displaystyle a)\) Sophie and Bertie are playing a game that involves breaking a bar of chocolate into pieces. The chocolate bar consists of \(\displaystyle 10\times 5\) squares. They take turns in splitting the chocolate along the dividing lines. The player who first breaks off a single square will lose the game. In each move, they are only allowed to touch and split one piece. Sophie starts the game. Can she make sure that she will win, whatever Bertie's moves are?

\(\displaystyle b)\) After the first game, Bertie wants to strike back. He wants Sophie to start again, but with a different rule: the player who first breaks off a single square will win the game. Can Sophie make sure that she will win again?

(6 pont)

solution (in Hungarian), statistics


K. 685. Steve went picking mushrooms. Since he is becoming better and better at spotting mushrooms, this time he found 62 penny bun mushrooms. The average number in his previous mushroom picking trips had been 30, which was thus increased to 32. How many penny bun mushrooms should he have found in order to increase the mean to 33?

(6 pont)

solution (in Hungarian), statistics


K. 686. Each of the integers 1 to 100 is written on a separate piece of paper. 20 pieces of paper are drawn at random from the 100 pieces. Show that it is always possible to select four out of the 20 such that the sum of two numbers equals the sum of the other two.

(6 pont)

solution (in Hungarian), statistics


K. 687. There are some toy robots waiting on one side of a street. In each move, it is allowed to instruct exactly four robots to cross the street. For what number of robots is it possible to make all the robots end up on the other side?

(6 pont)

solution (in Hungarian), statistics


K. 688. \(\displaystyle a)\) Is it possible to form pairs out of the numbers \(\displaystyle 1, 2, 3, 4, \dots, 23, 24\), so that the sum of each pair should be a perfect square?

\(\displaystyle b)\) Is it possible to form pairs out of the numbers \(\displaystyle 1, 2, 3, 4, \dots, 21, 22\) so that the sum of each pair should be a perfect square?

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'C'

Deadline expired on March 10, 2021.


C. 1651. The terms of a number sequence are generated as follows: the first term is \(\displaystyle 895\), and the following term is always obtained by multiplying the sum of the digits of the previous term by \(\displaystyle 61\). Determine the \(\displaystyle 2021\)st term of the sequence, and the sum of the first \(\displaystyle 2021\) terms.

(5 pont)

solution (in Hungarian), statistics


C. 1652. The shorter leg of each of two right-angled triangles has unit length. In each triangle, the right-angled vertex is at a unit distance from a point dividing the hypotenuse in a \(\displaystyle 2:1\) ratio: in one case it is the point closer to the right-angled vertex, and in the other case it is the point farther away. Prove that it is possible to select three out of the non-unit sides of the two triangles such that the three lengths form a right-angled triangle.

(5 pont)

solution (in Hungarian), statistics


C. 1653. How many solutions does the inequality

\(\displaystyle |x|+|y|<2021 \)

have over the set of pairs of integers?

(5 pont)

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C. 1654. Find the radius of each circle that is tangent to the graphs of the functions \(\displaystyle f(x)=\frac{3x-6}{4}\) and \(\displaystyle g(x)=\frac{28-4x}{3}\), and also touches the \(\displaystyle x\)-axis.

(5 pont)

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C. 1655. Solve the equation \(\displaystyle 2{(x+y-1831)}^2=(2x-1802)(2y-1860)\) over the set of pairs of real numbers.

(5 pont)

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C. 1656. Three consecutive terms of an arithmetic sequence are prime numbers greater than 3. Show that the common difference of the sequence is divisible by 3.

Proposed by L. Németh, Fonyód

(5 pont)

solution (in Hungarian), statistics


C. 1657. \(\displaystyle BCD\) and \(\displaystyle CAE\) are regular triangles drawn on legs \(\displaystyle BC\) and \(\displaystyle CA\) of a right-angled triangle \(\displaystyle ABC\), on the outside. Prove that the midpoints of the line segments \(\displaystyle AB\), \(\displaystyle CD\) and \(\displaystyle CE\) also form a regular triangle.

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on March 10, 2021.


B. 5150. Prove that there are only a finite number of positive integers that cannot be obtained by adding one of the digits of a smaller number to that number. What is the largest of these finite number of integers?

(4 pont)

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B. 5151. Prove that if \(\displaystyle a^2=b^2+ac=c^2+ab\), then two of the numbers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) are equal.

(3 pont)

solution (in Hungarian), statistics


B. 5152. Determine those positive integers greater than \(\displaystyle 1\) whose positive divisors can all be written on the circumference of a circle so that the ratio of every adjacent pair should be a prime.

Proposed by D. Lenger, Budapest and G. Szűcs, Szikszó

(5 pont)

solution (in Hungarian), statistics


B. 5153. Let \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\) denote the vertices of an equilateral triangle of unit side, and let \(\displaystyle D\) be a point on the extension of side \(\displaystyle AB\) beyond \(\displaystyle B\). The perpendicular drawn to line segment \(\displaystyle BC\) at \(\displaystyle B\) intersects line segment \(\displaystyle CD\) at \(\displaystyle E\) (see figure). Find the length of \(\displaystyle CE\), given that \(\displaystyle ED=1\).

Proposed by L. Szilassi and T. Tarcsay, Szeged

(4 pont)

solution (in Hungarian), statistics


B. 5154. Find all functions \(\displaystyle f\) taking on positive integer values, and defined on the set of positive integers such that \(\displaystyle f\big(f(n)\big)=2n\) and \(\displaystyle f(4n-3)=4n-1\) for all positive integers \(\displaystyle n\).

Proposed by S. Róka, Nyíregyháza

(4 pont)

solution (in Hungarian), statistics


B. 5155. The convex quadrilateral \(\displaystyle ABCD\) has no parallel sides, and the intersection of lines \(\displaystyle AB\) and \(\displaystyle CD\) is \(\displaystyle M\). Point \(\displaystyle X\) is moving along the interior of side \(\displaystyle AB\), and point \(\displaystyle Y\) is moving along the interior of side \(\displaystyle CD\) so that the equality \(\displaystyle AX:XB=DY:YC\) remains true. Show that the circles \(\displaystyle MXY\) all have another common point, different from \(\displaystyle M\).

(5 pont)

solution (in Hungarian), statistics


B. 5156. Let \(\displaystyle K\) be a convex polygon of \(\displaystyle 2n\) vertices where all sides have unit length and the opposite sides are parallel. Show that \(\displaystyle K\) can be dissected into a finite number of rhombuses of unit sides. How many rhombuses may there be?

(6 pont)

solution (in Hungarian), statistics


B. 5157. There are some integers on the blackboard. Each of three students (independently of each other) selects a number from the board at random, and writes it down in their notebook. Prove that the probability that the sum of the three numbers written down is divisible by 3 is at least \(\displaystyle 1/4\).

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on March 10, 2021.


A. 793. In the 43 dimension Euclidean space the convex hull of finite set \(\displaystyle S\) contains polyhedron \(\displaystyle P\). We know that \(\displaystyle P\) has 47 vertices. Prove that it is possible to choose at most 2021 points in \(\displaystyle S\) such that the convex hull of these points also contain \(\displaystyle P\), and this is sharp.

Submitted by Dömötör Pálvölgyi, Budapest

(7 pont)

statistics


A. 794. A polyomino \(\displaystyle P\) occupies \(\displaystyle n\) cells of an infinite grid of unit squares. In each move, we lift \(\displaystyle P\) off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have \(\displaystyle n-1\) cells in common. We say that \(\displaystyle P\) is a caterpillar of area \(\displaystyle n\) if, by means of a series of moves, we can free up all cells initially occupied by \(\displaystyle P\).

How many caterpillars of area \(\displaystyle 10^6 + 1\) are there?

Submitted by Nikolai Beluhov, Bulgaria

(7 pont)

statistics


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