KöMaL Problems in Mathematics, November 2020

Problems with sign 'K' Deadline expired on December 10, 2020.

K. 669. Let us consider the set of 3-digit positive integers containing all the digits 1, 2, 3 exactly once. Find the smallest positive integer that contains each number from the previous set as consecutive digits.

(6 pont)

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K. 670. Grandma bought two candles: the red candle was 2 cm longer than the blue one. On All Saints' Day she lit the red candle at 5:30 p.m. then she lit the blue candle at 7 p.m. and let them burn all the way down. The two candles were equal in length at 9:30 p.m. The blue candle burned out at 11 p.m and the red one finished at 11:30 p.m. What was the initial length of the red candle?

(6 pont)

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K. 671. We know that the first five terms of an increasing arithmetic sequence are all positive primes. Find the smallest prime at the \(\displaystyle 5^\text{th}\) position.

(6 pont)

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K. 672. A garden is divided into 16 patches as shown in the figure. In each patch, either roses or tulips or daisies or gerberas are grown: only one type of flower in each, but every row, every column, and both diagonals contain every type of flower. In how many different ways is it possible to arrange the flowers in this way? (Two arrangements are considered different if there exists a patch that contains a different kind of flower.)

(6 pont)

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K. 673. The students in a class (we do not know how many of them there are) decided that everyone would buy some small present to everyone else for Christmas, and they would also buy some present together for each of their 11 teachers. Unfortunately, the Christmas party was cancelled. Then they decided to divide the presents equally among all the siblings of the students. (Each sibling gets the same present.) Was that possible if the total number of siblings was 15?

(6 pont)

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Problems with sign 'C' Deadline expired on December 10, 2020.

C. 1630. The numbers 1 to 32 are written in the white fields of a chessboard, using each number once. Then the sum of the numbers in the adjacent fields is entered in each black field. What are the smallest and largest possible values of the sum of the numbers in the black fields?

(5 pont)

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C. 1631. Let \(\displaystyle AB\) be a chord in a unit circle. Triangle \(\displaystyle ABC\) is right-angled at \(\displaystyle B\), and vertex \(\displaystyle C\) lies on the circle. Triangle \(\displaystyle ABD\) is isosceles right-angled, and \(\displaystyle AB\) is the hypotenuse. How long is the chord \(\displaystyle AB\) if the two triangles have equal areas? What is this area?

(5 pont)

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C. 1632. How many different infinite arithmetic sequences of positive integer terms are there in which 24, 744 and 2844 all occur? Two arithmetic sequences are considered different if they have different first terms or different common differences.

(5 pont)

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C. 1633. Let \(\displaystyle P\) be an interior point of one side of a unit square. Consider all parallelograms with one vertex at \(\displaystyle P\), and one on each side of the square. Prove that if \(\displaystyle P\) is not the midpoint of the side then \(\displaystyle (i)\) there are exactly two rectangles among these parallelograms, and \(\displaystyle (ii)\) the sum of the areas of these two rectangles is 1.

(5 pont)

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C. 1634. Prove the following inequality:

\(\displaystyle \frac{1}{4} +\frac{1}{28} +\frac{1}{70} +\cdots +\frac{1}{(3k-2)(3k+1)} +\cdots +\frac{1}{2017\cdot 2020} < \frac{1}{3}. \)

(5 pont)

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C. 1635. Given two intersecting circles, construct\footnote[1]with straight edge and compasses on paper, or with appropriate geometric construction software a secant through one of the intersection points such that the segment bounded by the two circles is divided \(\displaystyle 1:2\) by the intersection point. Write down and explain the steps of the construction. (Elementary steps like bisecting an angle or reflecting a point in a line do not need to be described in detail.)

(5 pont)

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C. 1636. The Hungarian poet Dezső Kosztolányi spent a few weeks in Paris when he was a student. When he was given for change a ten-centime coin not in circulation any more, he wanted to give it away. He did not succeed, which he explained to himself by the expression on his face revealing his intentions. Therefore he decided to get 9 valid ten-centime coins, mix them with the worthless coin in his pocket, and by not looking at them he pays with one of them in a shop. He continued doing so until he had a single coin in his pocket: the coin out of circulation. What is the probability of this?

(5 pont)

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Problems with sign 'B' Deadline expired on December 10, 2020.

B. 5126. Prove that if \(\displaystyle n\ge 3\), then there exist \(\displaystyle n\) distinct positive integers such that the sum of their reciprocals is 1.

(3 pont)

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B. 5127. Given a convex angle and a line segment of length \(\displaystyle k\), determine the locus of those points inside the angle through which there exists a line cutting off a triangle of perimeter \(\displaystyle k\) from the angle.

(4 pont)

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B. 5128. Find all pairs of relatively prime integers \(\displaystyle (x,y)\) such that \(\displaystyle x^2 + x = y^3 + y^2\).

Proposed by L. Surányi, Budapest

(4 pont)

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B. 5129. Two players are taking turns in selecting one of the coefficients \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) of the polynomial \(\displaystyle x^3+ax^2+bx+c\), and giving it some integer value of their choice. Prove that the starting player can achieve that (after the three steps) all three roots of the polynomial should be integers (i.e. that the polynomial can be expressed as a product of three polynomials of integer coefficients).

(3 pont)

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B. 5130. There are \(\displaystyle n\) points in the plane. We know that for any \(\displaystyle k\) points (\(\displaystyle k\ge 2\)), it is possible to select two of them with distance at most 1. Show that the points can be covered with \(\displaystyle k-1\) disks of unit radius.

(5 pont)

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B. 5131. Let \(\displaystyle H\) be an equilateral triangle of unit area, let \(\displaystyle O\) be a fixed point, and for any point \(\displaystyle P\) let \(\displaystyle H_P\) denote the triangle obtained from triangle \(\displaystyle H\) by a parallel shift with vector \(\displaystyle \overrightarrow{OP}\). Consider the set \(\displaystyle N\) of points \(\displaystyle P\) for which the area of the intersection \(\displaystyle H\cap H_P\) is at least \(\displaystyle 4/9\). What is the area of \(\displaystyle N\)?

Based on the idea of V. Vígh, Székkutas

(5 pont)

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B. 5132. How many different strings of 2021 letters can be made of letters A, B and C such that the number of A's is even and the number of B's is of the form \(\displaystyle 3k+2\)?

(6 pont)

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B. 5133. Given six points in the space, no four of which are coplanar, prove that they can be divided into two sets of three such that the two triangular plates spanned by the two sets of three points should intersect each other.

(6 pont)

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Problems with sign 'A' Deadline expired on December 10, 2020.

A. 786. In a convex set \(\displaystyle S\) that contains the origin it is possible to draw \(\displaystyle n\) disjoint unit circles such that viewing from the origin non of the unit circles blocks out a part of another (or a complete) unit circle. Prove that the area of \(\displaystyle S\) is at least \(\displaystyle n^2/100\).

Submitted by: Dömötör Pálvölgyi, Budapest

(7 pont)

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A. 787. Let \(\displaystyle p_n\) denote the \(\displaystyle n^{\text{th}}\) prime number and define \(\displaystyle a_n=\lfloor p_n \nu \rfloor\), where \(\displaystyle \nu\) is a positive irrational number. Is it possible that there exist only finitely many \(\displaystyle k\) such that \(\displaystyle \binom{2a_k}{a_k}\) is divisible by \(\displaystyle p_i^{10}\) for all \(\displaystyle i=1,2,\ldots, 2020\)?

Submitted by: Abhishek Jha, Delhi, India and Ayan Nath, Tezpur, India

(7 pont)

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A. 788. Solve the following system of equations:

\(\displaystyle x+\frac1{x^3}=2y, \qquad y+\frac1{y^3}=2z, \qquad z+\frac1{z^3}=2w, \qquad w+\frac1{w^3}=2x. \)

(7 pont)

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