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KöMaL Problems in Mathematics, February 2016

Please read the rules of the competition.


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Problems with sign 'K'

Deadline expired on 10 March 2016.


K. 493. Is it possible to write the numbers 1, 2, 3, 4, 5, 6, 7 and 8 on the vertices of a cube so that the sum of the numbers on the vertices of each face is a prime number?

(6 pont)

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K. 494. The perimeters of two triangles are the integers \(\displaystyle y\) and \(\displaystyle y + 1\). Each triangle has two sides of integer lengths of \(\displaystyle x\) and \(\displaystyle x + 1\). Given that the sum of the perimeters is 27, how long are the third sides?

(6 pont)

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K. 495. In a class of 30 students, there are 12 more girls than boys. A team of three members is to be selected that contains at least one boy and at least one girl. How many different teams are possible? (Two teams are considered different if there is at least one person who is a member of one team but is not a member of the other.)

(6 pont)

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K. 496. Seven friends decided to form clubs. Each club is to have three members, and any two clubs may have at most one member in common. Can they form seven clubs?

(6 pont)

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K. 497. In a right-angled triangle \(\displaystyle ABC\), \(\displaystyle BC = 5\) and \(\displaystyle AB = 12\). \(\displaystyle M\) is the intersection of hypotenuse \(\displaystyle AC\) with the arc of radius \(\displaystyle AB\) centred at \(\displaystyle A\), and \(\displaystyle N\) is the intersection of hypotenuse \(\displaystyle AC\) with the arc \(\displaystyle BC\) centred at \(\displaystyle C\). Determine the distance between points \(\displaystyle M\) and \(\displaystyle N\).

(6 pont)

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K. 498. A circle is divided into twelve equal arcs, and the points of division are joined as shown in the figure. Determine the proportions of the areas of the rhombuses formed.

(6 pont)

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Problems with sign 'C'

Deadline expired on 10 March 2016.


C. 1336. In how many different ways is it possible to dissect a \(\displaystyle 6\times6\) square into \(\displaystyle 1\times3\) rectangles?

(5 pont)

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C. 1337. Csongor's wife sewed a leather sheath decorated with 77 beads for his husband's mouth harp, and gave it to him as a birthday present. Csongor liked it so much that he decided to surprise every member of his heritage preservation mouth harp band with a sheath like his own. He presented the sheaths in the main yurt erected for the celebration of the winter solstice. The yurt had a maximum capacity of 50 persons. Since the beads were bought in packets of 100, 7 beads remained, so Csongor's wife decorated her traditional headdress with them. How many members are there in Csongor's mouth harp band?

(5 pont)

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C. 1338. Let \(\displaystyle D\) denote a point on base \(\displaystyle AB\), and let \(\displaystyle E\) denote a point on leg \(\displaystyle BC\) of an isosceles triangle \(\displaystyle ABC\) such that the triangles \(\displaystyle ACD\), \(\displaystyle CDE\), and \(\displaystyle BDE\) are all isosceles, and triangle \(\displaystyle BDE\) is similar to triangle \(\displaystyle ABC\). Determine the angles of each triangle.

(5 pont)

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C. 1339. In a tree nursery, trees are planted at the lattice points of a \(\displaystyle 30~\rm m\times 40~m\) square grid. The distance between adjacent trees along the grid lines is 1 metre. A meadow mouse goes for a morning walk around the tree grove. During his walk, his distance from the closest tree is 1 metre at every time instant. What is the total distance covered by the meadow mouse if he walks around without turning back?

(5 pont)

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C. 1340. Points \(\displaystyle P\), \(\displaystyle Q\), \(\displaystyle R\), \(\displaystyle S\) lie on sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DA\) of a rectangle \(\displaystyle ABCD\), respectively. Line segments \(\displaystyle PR\) and \(\displaystyle QS\) are perpendicular. Prove that the midpoints of line segments \(\displaystyle SP\), \(\displaystyle PQ\), \(\displaystyle QR\) and \(\displaystyle RS\) form a rectangle, which is similar to \(\displaystyle ABCD\).

(5 pont)

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C. 1341. Find the largest power of 2 that divides 2016!.

(5 pont)

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C. 1342. Determine the value of the parameter \(\displaystyle a\) such that the equation below has exactly two solutions: \(\displaystyle x^3 -a=\sqrt[3]{x+a}\,\). Find the solutions, too.

(5 pont)

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Problems with sign 'B'

Deadline expired on 10 March 2016.


B. 4768. Find all fractions in which the numerator and denominator are both two-digit numbers such that the second digit of the numerator equals the first digit of the denominator, and the value of the fraction stays the same if these identical digits are deleted.

Proposed by A. Velkeyné Gréczi, Ipolyszög

(3 pont)

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B. 4769. In which triangles are the points dividing the sides 2:1 all concyclic?

Based on the idea of J. Szoldatics, Budapest

(3 pont)

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B. 4770. What may be the last digit of a positive integer \(\displaystyle n\ge 3\) if \(\displaystyle n+n^{2} +\dots+n^{2n-3} - 4\) is a prime?

(4 pont)

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B. 4771. In an aeroplane, there are one hundred seats, booked by one hundred passengers, each having their assigned seat. However, the first passenger does not care, and sits down on a random seat. When the other passengers enter one by one, each of them tries to take his or her own seat, or, if that seat is already taken, selects another one at random. What is the probability that the hundredth passenger is able to take his own seat?

Proposed by N. Nagy, Budapest

(5 pont)

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B. 4772. Is it true that if the four sides in two convex quadrilaterals are pairwise equal, and the two diagonals are also pairwise equal then the two quadrilaterals are congruent?

Proposed by V. Vígh, Szeged

(4 pont)

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B. 4773. The centre of the inscribed circle of triangle \(\displaystyle ABC\) is \(\displaystyle O\), and the centre of the circumscribed circle is \(\displaystyle K\). Prove that the vector \(\displaystyle \frac{\overrightarrow{AB}}{AB}+ \frac{\overrightarrow{BC}}{BC}+ \frac{\overrightarrow{CA}}{CA}\) is perpendicular to line \(\displaystyle OK\).

Proposed by G. Holló, Budapest

(5 pont)

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B. 4774. The parabolas \(\displaystyle p_1\) \(\displaystyle \big(y=-x^2+b_1 x+c_1\big)\) and \(\displaystyle p_2\) \(\displaystyle \big(y=-x^2+b_2 x+c_2\big)\) are tangent to the parabola \(\displaystyle p_3\) \(\displaystyle \big(y=x^2+b_3x+c_3\big)\). Prove that the line connecting the points of tangency is parallel to the common tangent of \(\displaystyle p_1\) and \(\displaystyle p_2\).

Kvant

(5 pont)

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B. 4775. Find those pairs \(\displaystyle (n,k)\) of positive integers for which

\(\displaystyle \sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}^{n}\ge \bigg(\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}\bigg)^{\!\!n} \)

for all real numbers \(\displaystyle a_1\ge a_2\ge \dots \ge a_{2k+1}\ge 0\).

Proposed by Á. Somogyi, Budapest

(6 pont)

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B. 4776. Let \(\displaystyle \mathcal{O}\) be a regular octahedron. How many different axes of rotation are there such that \(\displaystyle \mathcal{O}\) is mapped onto itself by a rotation through an angle of at most \(\displaystyle 180^{\circ}\)?

(6 pont)

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Problems with sign 'A'

Deadline expired on 10 March 2016.


A. 662. The points \(\displaystyle A_1\), \(\displaystyle A_2\), \(\displaystyle A_3\), \(\displaystyle A_4\), \(\displaystyle B_1\), \(\displaystyle B_2\), \(\displaystyle B_3\), \(\displaystyle B_4\) lie on a parabola in this order. For every pair \(\displaystyle (i,j)\) with \(\displaystyle 1\le i,j\le4\) and \(\displaystyle i\ne j\), let \(\displaystyle r_{ij}\) denote the ratio in which the line \(\displaystyle A_jB_j\) divides the segment \(\displaystyle A_iB_i\). (That is, if \(\displaystyle A_iB_i\) and \(\displaystyle A_jB_j\) meet at \(\displaystyle X\) then \(\displaystyle r_{ij}=\frac{A_iX}{XB_i}\).) Show that if two of the numbers \(\displaystyle r_{12} \cdot r_{21} \cdot r_{34} \cdot r_{43}\), \(\displaystyle r_{13} \cdot r_{31} \cdot r_{24} \cdot r_{42}\) and \(\displaystyle r_{14} \cdot r_{41} \cdot r_{23} \cdot r_{32}\) coincide then the third one is also equal to them.

(5 pont)

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A. 663. There are given two positive integers: \(\displaystyle k\) and \(\displaystyle \ell\). A square with horizontal and vertical sides is divided into finitely many rectangles by line segments such that the following statements are satisfied: \(\displaystyle (i)\) every horizontal or vertical line of the plane contains at most one of the segments; \(\displaystyle (ii)\) no two segments cross each other in their interiors; \(\displaystyle (iii)\) every horizontal line, intersecting the square but not containing any of the segments, intersects exactly \(\displaystyle k\) rectangles; \(\displaystyle (iv)\) every vertical line, intersecting the square but not containing any of the segments, intersects exactly \(\displaystyle \ell\) rectangles. What can be the number of rectangles?

Russian problem

(5 pont)

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A. 664. Let \(\displaystyle a_1<a_2<\ldots<a_n\) be an arithmetic progression of positive integers. Prove that \(\displaystyle [a_1,a_2,\ldots,a_n] \ge [1,2,\ldots,n]\). (The symbol \(\displaystyle [\ldots]\) stands for the least common multiple.)

(5 pont)

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