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Problems in Mathematics, October 2014

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 November 2014.

A. 623. Let \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) be three distinct positive reals. The logarithmic mean of \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) is defined by

\(\displaystyle L(a,b,c) = 2\left( \frac{a}{(\ln a-\ln b)(\ln a-\ln c)} + \frac{b}{(\ln b-\ln c)(\ln b-\ln a)} + \frac{c}{(\ln c-\ln a)(\ln c-\ln b)} \right).\)

Prove that \(\displaystyle \sqrt[3]{abc} < L(a,b,c) < \frac{a+b+c}{3}\).

(5 points)

Solution, statistics

A. 624. \(\displaystyle a)\) Prove that for every infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) there exists some \(\displaystyle C>0\) such that for every positive integer \(\displaystyle r\) there are positive integers \(\displaystyle n\), \(\displaystyle m\) satisfying \(\displaystyle |n-m|\ge r\) and \(\displaystyle |x_n-x_m|<\frac{C}{|n-m|}\).

\(\displaystyle b)\) Show that for every \(\displaystyle C>0\) there exists an infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) and a positive integer \(\displaystyle r\) such that \(\displaystyle |x_n-x_m|>\frac{C}{|n-m|}\) holds true for every pair \(\displaystyle n\), \(\displaystyle m\) of positive integers with \(\displaystyle |n-m|\ge r\).

(CIIM6, Costa Rica)

(5 points)

Statistics

A. 625. Let \(\displaystyle n\ge2\), and let \(\displaystyle \mathcal{S}\) be a family of some subsets of \(\displaystyle \{1,2,\ldots,n\}\) with the property that \(\displaystyle |A\cup B\cup C\cup D|\le n-2\) for all \(\displaystyle A,B,C,D\in\mathcal{S}\). Show that \(\displaystyle |\mathcal{S}|\le 2^{n-2}\).

(CIIM6, Costa Rica)

(5 points)

Statistics


Problems with sign 'B'

Deadline expired on 10 November 2014.

B. 4651. A positive integer \(\displaystyle n\) is said to be exotic if it is divisible by the number of its positive factors. Prove the following statements:

\(\displaystyle a)\) If an exotic number is odd then it is a perfect square.

\(\displaystyle b)\) There are infinitely many exotic numbers.

(3 points)

Statistics

B. 4652. The angles of a triangle are \(\displaystyle \alpha\), \(\displaystyle \beta\) and \(\displaystyle \gamma\). Determine the angles of the triangle formed by the tangents drawn to the circumscribed circle at the vertices.

(3 points)

Statistics

B. 4653. How many ordered triples of positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) are there for which \(\displaystyle [a,b,c]=10!\) and \(\displaystyle (a,b,c)=1\)? (\(\displaystyle (a,b,c)\) denotes the greatest common divisor, and \(\displaystyle [a,b,c]\) denotes the least common multiple.)

(4 points)

Statistics

B. 4654. In a triangle \(\displaystyle ABC\), let \(\displaystyle AD\) be an altitude, let \(\displaystyle BE\) be an angle bisector, and let \(\displaystyle CF\) be a median. Prove that the lines \(\displaystyle AD\), \(\displaystyle BE\) and \(\displaystyle CF\) are concurrent exactly if \(\displaystyle ED\) is parallel to \(\displaystyle AB\).

(4 points)

Statistics

B. 4655. Is there a solution of the equation

\(\displaystyle 2012^{2015}=\binom n2 +\binom k2\)

on the set of positive integers?

Suggested by B. Maga

(5 points)

Statistics

B. 4656. Prove that any region of space bounded by the planes of the lateral faces of a four-sided convex pyramid can be intersected with a plane such that the intersection is a parallelogram.

(4 points)

Statistics

B. 4657. The radius of the inscribed circle of a triangle is \(\displaystyle r\), and the radius of the circumscribed circle is \(\displaystyle R\). Assume that \(\displaystyle R < r \big(\sqrt{2}+1\big)\). Does this condition imply that the triangle is acute-angled?

Suggested by T. Káspári, Paks

(5 points)

Statistics

B. 4658. Solve the equation

\(\displaystyle 8^{2x-1}-1=343^{x-1}+\frac{3}{14} 28^x.\)

(6 points)

Statistics

B. 4659. Given \(\displaystyle n\) points in a unit circle, what is the maximum possible value of the product of the distances of all point pairs?

(6 points)

Statistics


Problems with sign 'C'

Deadline expired on 10 November 2014.

C. 1245. Consider two consecutive triangular numbers and add one of them to the triple of the other. Prove that the result is also a triangular number.

(5 points)

This problem is for grade 1 - 10 students only.

Statistics

C. 1246. The height of a cyclic trapezium is 30 cm, its legs are 34 cm. The trapezium has an inscribed circle. Determine the distance between the points of tangency on the legs.

(5 points)

This problem is for grade 1 - 10 students only.

Statistics

C. 1247. Solve the following simultaneous equations:

\(\displaystyle x+\sqrt y =1,\)

\(\displaystyle \sqrt x+y =1.\)

(5 points)

Statistics

C. 1248. Determine all three-digit integers \(\displaystyle \overline{abc}\) such that \(\displaystyle \overline{abc}=a!+b!+c!\), where \(\displaystyle n!\) denotes the product of positive integer 1 to \(\displaystyle n\).

(5 points)

Statistics

C. 1249. Solve the following equation:

\(\displaystyle \sqrt{\frac{\cos {15}^\circ}2 x^2 - \cos {45}^\circ x + \sin {15}^\circ}=3 +4\sin^2 15^\circ.\)

(5 points)

Statistics

C. 1250. The sides of a triangle are \(\displaystyle a=2t-1\), \(\displaystyle b=t^2-1\), \(\displaystyle c=t^2-t+1\), where \(\displaystyle t>1\) is a real number. Prove that the radius of the inscribed circle of the triangle is \(\displaystyle (t-1)\frac{\sqrt 3}2\).

(5 points)

This problem is for grade 11 - 12 students only.

Statistics

C. 1251. Three circular segments are cut off a semicircular metal plate to form a trapezium. What should be the dimensions of the trapezium to minimize the area cut off?

(5 points)

This problem is for grade 11 - 12 students only.

Statistics


Problems with sign 'K'

Deadline expired on 10 November 2014.

K. 427. In how many different ways is it possible to select positive integers, each consisting of digits of 8 only such that their sum is 1000?

(6 points)

This problem is for grade 9 students only.

Statistics

K. 428. I have chosen a positive integer. The sum of all odd numbers up to my number is 2014 greater than the sum of all the even numbers. What is my number?

(6 points)

This problem is for grade 9 students only.

Statistics

K. 429. The measure of the angle lying at vertex \(\displaystyle C\) of an isosceles triangle \(\displaystyle ABC\) is \(\displaystyle 120^\circ\). The perpendicular bisectors of the legs intersect the base at the points \(\displaystyle D\) and \(\displaystyle E\). Show that the area of triangle \(\displaystyle ABC\) is three times the area of triangle \(\displaystyle CDE\).

(6 points)

This problem is for grade 9 students only.

Statistics

K. 430. Consider the notation \(\displaystyle 33335 = 3_{4}5_{1}\), that is, the number in the subscript denotes the number of identical digits next to each other (the numbers in the subscripts must be positive integers). Determine the values of the letters in the expression \(\displaystyle 1_{x}4_{y}3_{z}8_{w}+ 4_{p}8_{q}3_{r} = 5_{2}9_{3} 7_{3}2_{2} 1_{1}\).

(6 points)

This problem is for grade 9 students only.

Statistics

K. 431. A circular rubber band of radius 2 cm is placed between two parallel sticks lying on the table. The sticks are moved towards each other until their separation is 2 cm, compressing the rubber ring between them. The resulting figure consists of two semicircles and two parallel line segments. The length of the rubber band doesn't changed. By what factor did the area enclosed by the rubber band change in the ``compression''?

(6 points)

This problem is for grade 9 students only.

Statistics

K. 432. The lengths of the legs of a right-angled triangle are \(\displaystyle a\) and \(\displaystyle b\), the hypotenuse is \(\displaystyle c\). Given that \(\displaystyle a+b=4+c\), what is the relationship between the measures of the perimeter and area of the triangle?

(6 points)

This problem is for grade 9 students only.

Statistics


Send your solutions to the following address:

    KöMaL Szerkesztőség (KöMaL feladatok),
    Budapest 112, Pf. 32. 1518, Hungary
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