KöMaL Problems in Mathematics, October 2015
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Problems with sign 'K'Deadline expired on November 10, 2015. 
K. 469. A certain paint needs to be diluted in a \(\displaystyle 2 : 1.5\) ratio, that is, 2 litres of water need to be added to 1.5 litres of paint. Violette Palette, the artist first made 9 litres of a mixture, half paint, half water. Then she realized that this was the wrong ratio, and calculated how much more water to add to achieve the correct proportion. However, instead of the amount of water needed, she added an equal amount of paint by mistake. The second time she did not make any mistake, and added the appropriate quantity of water required for the correct ratio. How many litres of mixture did she get eventually?
(6 pont)
K. 470. We have cubes of two different sizes, each with edges of integer length in cm. The edges of the red cubes are 5 cm longer than the edges of the blue ones. By stacking 15 cubes on top of each other, we got a tower of height 140 cm. How long are the edges if the difference between the numbers of red and blue cubes used is as small as possible?
(6 pont)
K. 471. Barbara has one 5forint coin (HUF, Hungarian currency), one 10forint coin, one 20forint coin, three 50forint coins and three 100forint coins in her purse. How many different amounts can she pay exactly (that is, without getting back any change)?
(6 pont)
K. 472. Find the sum of all positive twodigit numbers with exactly 12 divisors.
(6 pont)
K. 473. What is the sum of the digits in the binary (base2) representation of the product \(\displaystyle 2^{2015}\cdot 15\)?
(6 pont)
K. 474. Ann and Bob are playing a word guessing game. Anna thinks of a meaningful Hungarian word of four letters, which Bob is trying to guess. If Bob tries a certain Hungarian word of four letters, Ann will tell him how many of its letters occur in her word, too, and how many of those are in the correct place and how many are in the wrong place. What may have been the word that Ann had in mind? (No knowledge of the Hungarian language is required. Note that O and Ó are different letters in the Hungarian alphabet.)

(6 pont)
Problems with sign 'C'Deadline expired on November 10, 2015. 
C. 1308. If appropriate triangles are cut into two parts with a line through the vertex with the largest angle, two isosceles triangles are obtained. What may be the angles of an obtuseangled triangle if this division into two parts can be accomplished in two different ways?
(5 pont)
C. 1309. \(\displaystyle t\) denotes the area of a certain triangle, \(\displaystyle R\) is the radius of the circumscribed circle, and \(\displaystyle r\) is the radius of the incircle. Prove that \(\displaystyle \frac{t}{3}<Rr\).
(5 pont)
C. 1310. Alex took \(\displaystyle 19\,500\) forints (HUF, Hungarian currency) with him on a fourday trip. On each day, he spent one third of his remaining money plus a constant amount. What was the constant amount if his money lasted just until the end of the trip?
(5 pont)
C. 1311. Each of the numbers \(\displaystyle \frac{1}{3}\); 0,375; 1; 1,4; \(\displaystyle \sqrt{2}\); \(\displaystyle \frac{13}{8}\); 2; \(\displaystyle \frac{13}{5}\); \(\displaystyle \frac{8}{3}\); 3; 4; \(\displaystyle \sqrt{18}\); \(\displaystyle \sqrt{32}\) is given either a plus sign or a minus sign, and then the sum is calculated. In how many different ways may we choose the signs to get \(\displaystyle 1\) as a sum?
(5 pont)
C. 1312. Suppose that \(\displaystyle xy+x+y=44\) and \(\displaystyle x^2y+xy^2=448\). Then evaluate \(\displaystyle x^2+y^2\).
M&IQ
(5 pont)
C. 1313. One vertex of an isosceles triangle is the point \(\displaystyle (0,1)\). One of the other two vertices lies on the \(\displaystyle x\)axis, and the other lies on the line of equation \(\displaystyle y=3\). What is the area of the triangle?
(5 pont)
C. 1314. Two sides of a triangle are of unit length and they enclose an angle of \(\displaystyle 108^{\circ}\). Inscribe a regular pentagon in the triangle such that three sides of the pentagon lie on the sides of the triangle. How long are the sides of the inscribed pentagon?
(5 pont)
Problems with sign 'B'Deadline expired on November 10, 2015. 
B. 4732. The teacher of a class of 36 students enters the averages of the mathematics test grades in a \(\displaystyle 6 \times 6\) table. Each student has a different mean grade. The teacher marks the largest entry in each column of the table. He finds that all of the 6 marked numbers lie in different rows. Then he marks the largest entry in each row, and finds that these all lie in different columns. Prove that the two sets of six numbers are equal.
Proposed by J. Szoldatics, Budapest
(3 pont)
B. 4733. Every edge of a simple connected graph of \(\displaystyle n\ge 2\) vertices is labelled with either a 1 or with a 2. Then each vertex is assigned with the product of the numbers on the edges product of the numbers on the related edges on it. Show that there will be a pair of two vertices assigned with the same number.
Proposed by A. Hujdurović, Koper
(3 pont)
B. 4734. Some fields (unit cubes) constituting a cubical lattice of edge 2015 units are infected by an unknown disease. The disease will spread if at least \(\displaystyle t\) fields in some row parallel to any edge of the cube are infected \(\displaystyle (1 \le t \le 2015)\). In that case, every field of that row will become infected in one minute. How many fields need to be infected initially in order to
\(\displaystyle a)\) make it possible
\(\displaystyle b)\) be certain
that the infection reaches all fields of the cube?
Proposed by G. Mészáros, Budapest
(6 pont)
B. 4735. Construct a cyclic quadrilateral, given one vertex, the line of the diagonal through the vertex, and the intersections of the lines of the two pairs of opposite sides.
(4 pont)
B. 4736. Let \(\displaystyle n\) be a positive integer. Solve the simultaneous equations
\(\displaystyle \sum_{i=1}^{n} {x_i} = \sum_{i=1}^{n} \bigx_i^3\big = \sum_{i=1}^{n} {\frac{2 {x_i}^3}{x_i^2+1}}. \)
Proposed by K. Williams, Szeged
(5 pont)
B. 4737. \(\displaystyle D\) is the foot of the altitude drawn to the hypotenuse \(\displaystyle AB\) of a rightangled triangle \(\displaystyle ABC\). The angles bisectors of \(\displaystyle \angle ACD\) and \(\displaystyle \angle BCD\) intersect hypotenuse \(\displaystyle AB\) at \(\displaystyle E\) and \(\displaystyle F\), respectively. Determine the ratio of the inradius of triangle \(\displaystyle ABC\) to the circmradius of triangle \(\displaystyle CEF\).
Proposed by B. Bíró, Eger
(5 pont)
B. 4738. \(\displaystyle C\) is an arbitrary point of a circle \(\displaystyle k\) of diameter \(\displaystyle AB\), different from \(\displaystyle A\) and \(\displaystyle B\). Drop a perpendicular from \(\displaystyle C\) onto diameter \(\displaystyle AB\). The foot of the perpendicular on line segment \(\displaystyle AB\) is \(\displaystyle D\), and the other intersection with the circle \(\displaystyle k\) is \(\displaystyle E\). The circle of radius \(\displaystyle CD\) centred at \(\displaystyle C\) intersects circle \(\displaystyle k\) at points \(\displaystyle P\) and \(\displaystyle Q\). Let \(\displaystyle M\) denote the intersection of line segments \(\displaystyle CE\) and \(\displaystyle PQ\). Dertermine the value of \(\displaystyle \frac{PM}{PE} + \frac{QM}{QE}\).
Proposed by B. Bíró, Eger
(4 pont)
B. 4739. Consider all real numbers \(\displaystyle x\) for which \(\displaystyle \tan x + \cot x\) is a positive integer. Find those of them for which \(\displaystyle \tan^3x + \cot^3 x\) is a prime number.
Proposed by B. Bíró, Eger
(4 pont)
B. 4740. Eight unit cubes, with corresponding edges parallel, are glued together to form a solid. Prove that the surface area of the resulting solid is at least 24 units. Only the outer surface counts, even if the resulting solid contains a cavity.
(6 pont)
Problems with sign 'A'Deadline expired on November 10, 2015. 
A. 650. There is given an acuteangled triangle \(\displaystyle ABC\) with a point \(\displaystyle X\) marked on its altitude starting from \(\displaystyle C\). Let \(\displaystyle D\) and \(\displaystyle E\) be the points on the line \(\displaystyle AB\) that satisfy \(\displaystyle \sphericalangle DCB=\sphericalangle ACE=90^\circ\). Let \(\displaystyle K\) and \(\displaystyle L\) be the points on line segments \(\displaystyle DX\) and \(\displaystyle EX\), respectively, such that \(\displaystyle BK=BC\) and \(\displaystyle AL=AC\). Let the line \(\displaystyle AL\) meet \(\displaystyle BK\) and \(\displaystyle BC\) at \(\displaystyle Q\) and \(\displaystyle R\), respectively; finally let the line \(\displaystyle BK\) meet \(\displaystyle AC\) at \(\displaystyle P\). Show that the quadrilateral \(\displaystyle CPQR\) has an inscribed circle.
(5 pont)
A. 651. Determine all real polynomials \(\displaystyle P(x)\) that satisfy
\(\displaystyle P\big(x^32\big)=P{(x)}^32. \)
CIIM 2015, Mexico
(5 pont)
A. 652. Prove that there exists a real number \(\displaystyle C>1\) with the following property: whenever \(\displaystyle n>1\) and \(\displaystyle a_0<a_1<\cdots<a_n\) are positive integers such that \(\displaystyle \frac1{a_0},\frac1{a_1},\ldots,\frac1{a_n}\) form an arithmetic progression, then \(\displaystyle a_0> C^n\).
CIIM 2015, Mexico
(5 pont)
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