KöMaL Problems in Mathematics, September 2019

Problems with sign 'K' Deadline expired on October 10, 2019.

K. 624. The integers 0 to 9 are arranged along a straight line in some order.

\(\displaystyle a)\) Find a possible arrangement in which the sum of any three adjacent numbers is less than 15.

\(\displaystyle b)\) Is there an arrangement of this kind if 0 is not included in the numbers?

(6 pont)

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K. 625. How many six-digit numbers are there in which each digit occurs exactly as many times as its value?

(6 pont)

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K. 626. The table below shows the statistics of a round-robin football championship of four participating teams. Teams are listed in alphabetical order of their names. Every team played every other team exactly once. The winner of each game scored 3 points, the losing team scored 0, and in a draw each team scored 1 point.

Given that the result of the Left Foot–Right Foot game was 3-1, and that the Headers scored a goal in every game they played, determine the result of each individual game.

(6 pont)

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K. 627. The teacher selects a student from a class at random. The probability of selecting a boy is \(\displaystyle 2/3\) of the probability that he selects a girl. What fraction of the whole class are girls?

(6 pont)

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K. 628. Four identical rectangular sheets of paper are placed on the table to form a larger rectangle without gaps or overlaps. The area of the resulting rectangle is \(\displaystyle 1200~\mathrm{cm}^{2}\). Given that it is not possible to transfer any rectangle to any other only by translation, find the perimeter of the large rectangle.

(6 pont)

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Problems with sign 'C' Deadline expired on October 10, 2019.

C. 1553. Determine the constant term of the expression \(\displaystyle \left(x^{12}+\frac1{x^{18}}\right)^{25}\).

(5 pont)

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C. 1554. One side of a rectangle is \(\displaystyle \frac{1+\sqrt5}{2}\) times as long as the other side. The rectangle is dissected and put together to form a square of the same area. What is the ratio of the diagonal of the rectangle to the diagonal of the square?

(5 pont)

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C. 1555. Solve the equation

\(\displaystyle x+y^2=4z^2 \)

over the set of positive prime numbers.

(5 pont)

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C. 1556. The interior angle bisector drawn from vertex \(\displaystyle C\) of triangle \(\displaystyle ABC\) intersects the opposite side at point \(\displaystyle P\). The distance of point \(\displaystyle P\) from the sides is \(\displaystyle \frac{24}{11}\), and \(\displaystyle AC=6\), \(\displaystyle BC=5\). Find the length of side \(\displaystyle AB\).

(5 pont)

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C. 1557. Two numbers are selected at random from the set of two-digit positive integers. What is the probability that they will have a digit in common?

(5 pont)

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C. 1558. Depending on the value of the nonzero parameter \(\displaystyle a\), how many points do the circle \(\displaystyle x^2+y^2=1\) and the parabola \(\displaystyle y=ax^2-1\) have in common?

(5 pont)

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C. 1559. The base of a tetrahedron is a regular triangle, and the three lateral faces unfolded and laid on the plane form a trapezium with sides 10, 10, 10 and 14 units of length. Find the sum of the lengths of all edges of the tetrahedron, and also find its surface area.

(5 pont)

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Problems with sign 'B' Deadline expired on October 10, 2019.

B. 5038. Let \(\displaystyle P\) be a point in the interior of a regular octagon \(\displaystyle ABCDEFGH\). Show that the sum of the areas of triangles \(\displaystyle ABP\), \(\displaystyle CDP\), \(\displaystyle EFP\) and \(\displaystyle GHP\) equals the sum of the areas of triangles \(\displaystyle BCP\), \(\displaystyle DEP\), \(\displaystyle FGP\) and \(\displaystyle HAP\).

(3 pont)

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B. 5039. Every entry in a \(\displaystyle 2019\times 2019\) table is either \(\displaystyle (+1)\) or \(\displaystyle (-1)\). If the sum of each row and the sum of each column are calculated, how many different numbers may be obtained at most?

Proposed by I. Blahota, Nyíregyháza

(3 pont)

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B. 5040. In a square \(\displaystyle ABCD\), let \(\displaystyle F\) be an interior point of side \(\displaystyle AB\), and let \(\displaystyle E\) be an interior point of side \(\displaystyle AD\). Draw a perpendicular to line \(\displaystyle CE\) at point \(\displaystyle E\), and a perpendicular to line \(\displaystyle CF\) at point \(\displaystyle F\). Denote the intersection of the two perpendiculars by \(\displaystyle M\). Given that the area of triangle \(\displaystyle CEF\) is half the area of pentagon \(\displaystyle BCDEF\), prove that point \(\displaystyle M\) lies on diagonal \(\displaystyle AC\) of the square.

(4 pont)

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B. 5041. An \(\displaystyle n \times n\) table of real numbers in each field is called a zero square if the sum of the numbers in every \(\displaystyle 2 \times 2\) square part of it (therefore in the whole table, too) is zero. (The diagram shows a \(\displaystyle 3\times3\) example.)

What is the largest possible \(\displaystyle n\) for which there exists an \(\displaystyle n \times n\) zero square such that the entries are not all zeros?

(5 pont)

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B. 5042. The convex quadrilateral \(\displaystyle ABCD\) is not a trapezium, and diagonals \(\displaystyle AC\) and \(\displaystyle BD\) are equal in length. Let \(\displaystyle M\) denote the intersection of the diagonals. Show that the other intersection (different from \(\displaystyle M\)) of the circles \(\displaystyle ABM\) and \(\displaystyle CDM\) lies on the angle bisector of angle \(\displaystyle BMC\).

(4 pont)

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B. 5043. Prove that the set \(\displaystyle \{1,2,3,4,5,6,7,8,9,10,11,12,13\}\) has an odd number of nonempty subsets in which the arithmetic mean of the elements is an integer.

Proposed by S. Róka, Nyíregyháza

(5 pont)

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B. 5044. Let \(\displaystyle D\) be a point in the interior of side \(\displaystyle AB\) in triangle \(\displaystyle ABC\), and \(\displaystyle E\) be a point in the interior of side \(\displaystyle AC\). The intersection of line segments \(\displaystyle BE\) and \(\displaystyle CD\) is \(\displaystyle M\). Let \(\displaystyle x\) denote the area of triangle \(\displaystyle BCM\), and let \(\displaystyle y\) denote the area of triangle \(\displaystyle EDM\). Prove that \(\displaystyle T_{ABC}\ge x \frac{\sqrt x+\sqrt y}{\sqrt x-\sqrt y}\).

(6 pont)

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B. 5045. For which positive integers \(\displaystyle n\) is there an appropriate order \(\displaystyle a_1,a_2,\dots,a_n\) of the first \(\displaystyle n\) positive integers such that the numbers \(\displaystyle a_1+1,a_2+2,\dots,a_n+n\) are all perfect powers? (A number is called a perfect power if it can be represented in the form \(\displaystyle a^b\), where \(\displaystyle a, b\ge 2\) are integers.)

(6 pont)

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Problems with sign 'A' Deadline expired on October 10, 2019.

A. 755. Prove that every polygon that has a center of symmetry can be dissected into a square such that it is divided into finitely many polygonal pieces, and all the pieces can only be translated. (In other words, the original polygon can be divided into polygons \(\displaystyle A_1, A_2,\dots, A_n\), a square can be divided into polygons a \(\displaystyle B_1, B_2,\dots, B_n\) such that for \(\displaystyle 1\le i\le n\) polygon \(\displaystyle B_i\) is a translated copy of polygon \(\displaystyle A_i\).)

(7 pont)

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A. 756. Find all functions \(\displaystyle f\colon \mathbb{R}\to \mathbb{R}\) (functions with domain \(\displaystyle \mathbb{R}\) and values from \(\displaystyle \mathbb{R}\)) which satisfy the following two conditions:

\(\displaystyle (i)\) \(\displaystyle f(x+1)=f(x)+1\);

\(\displaystyle (ii)\) \(\displaystyle f(x^2)=\big(f(x)\big)^2\).

(Based on a problem of Romanian Masters of Mathematics)

(7 pont)

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A. 757. For every \(\displaystyle n\) non-negative integer let \(\displaystyle S(n)\) denote a subset of the positive integers, for which \(\displaystyle i\) is an element of \(\displaystyle S(n)\) if and only if the \(\displaystyle i\)-th digit (from the right) in the base two representation of \(\displaystyle n\) is a digit 1.

Two players, \(\displaystyle A\) and \(\displaystyle B\) play the following game: first, \(\displaystyle A\) chooses a positive integer \(\displaystyle k\), then \(\displaystyle B\) chooses a positive integer \(\displaystyle n\) for which \(\displaystyle 2^n\ge k\). Let \(\displaystyle X\) denote the set of integers \(\displaystyle \{0,1,\dots,2^n-1\}\), let \(\displaystyle Y\) denote the set of integers \(\displaystyle \{0,1,\dots,2^{n+1}-1\}\). The game consists of \(\displaystyle k\) rounds, and in each round player \(\displaystyle A\) chooses an element of set \(\displaystyle X\) or \(\displaystyle Y\), then player \(\displaystyle B\) chooses an element from the other set. For \(\displaystyle 1\le i\le k\) let \(\displaystyle x_i\) denote the element chosen from set \(\displaystyle X\), let \(\displaystyle y_i\) denote the element chosen from set \(\displaystyle Y\).

Player \(\displaystyle B\) wins the game, if for every \(\displaystyle 1\le i\le k\) and \(\displaystyle 1\le j\le k\) \(\displaystyle x_i<x_j\) if and only if \(\displaystyle y_i<y_j\) and \(\displaystyle S(x_i)\subset S(x_j)\) if and only if \(\displaystyle S(y_i)\subset S(y_j)\).

Which player has a winning strategy?

(Proposed by Levente Bodnár, Cambridge)

(7 pont)

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