Competitions Portal

Problems in Mathematics, April 2012

Please read the rules of the competition.

Problems sign 'A'

Deadline expired.

A. 560. Given a right circular cone with vertex O, and a fixed point P in the base plane of the cone. Draw a line x through P which intersects the base circle of the cone at two points, X₁ and X₂. Show that $\tan\frac{\angle POX_1}2 \cdot \tan\frac{\angle POX_2}2$ does not depend on the choice of the line x.

(5 points)

Solution, statistics

A. 561. Show that

$\frac{a^3b}{{(3a+b)}^p} + \frac{b^3c}{{(3b+c)}^p} +\frac{c^3a}{{(3c+a)}^p} \ge \frac{a^2bc}{{(2a+b+c)}^p} + \frac{b^2ca}{{(2b+c+a)}^p} + \frac{c^2ab}{{(2c+a+b)}^p}$

holds for all positive numbers a, b, c, p.

(5 points)

Solution, statistics

A. 562. Let f(k)=2^k+1 for arbitrary positive integer k. Is there any positive integer n which divides f(f(n)), but does not divide f(f(f(n)))?

(5 points)

Problems sign 'B'

Deadline expired.

B. 4442. Ann, Bill and Cecilia play the following game. They take turns choosing an integer between 1 and 10, and adding that number to the sum of the numbers that have been said so far. (For simplicity, in each turn, the player states the sum.) The player who is first able to say 100 wins the game. Prove that the two girls, with the appropriate strategy, are able to achieve that one of them is the winner.

Suggested by B. Futó, New York

(3 points)

Solution (in Hungarian)

B. 4443. a₁,a₂,...,a_n és b₁,b₂,...,b_k are two given sequences with positive integer terms with a_i $\le$ k and b_j $\le$ n. Show that there are integers 0 $\le$ i₁<i₂ $\le$ n and 0 $\le$ j₁<j₂ $\le$ k such that $a_{i_1+1}+\ldots +a_{i_2}=b_{j_1+1}+\dots +b_{j_2}$ .

(5 points)

Solution (in Hungarian)

B. 4444. The points E and G, respectively, are obtained by rotating vertex D of a square ABCD about A and vertex B (lying opposite D) about C outwards, through the same acute angle $\alpha$ . The intersection of lines AE and BG is F. Line GD intersects the circles GCB and BDE again at points L and M, respectively. Prove that the points A, B, F, L, M are cyclic.

Suggested by J. Bodnár

(3 points)

Solution (in Hungarian)

B. 4445. A convex solid has six square faces and eight regular hexagonal faces. Given that no two square faces have a vertex in common, and the solid has an edge of unit length, determine the volume of the solid.

(4 points)

Solution (in Hungarian)

B. 4446. How many squares are formed by the lattice points in an n×n square lattice?

(4 points)

Solution (in Hungarian)

B. 4447. Solve the equation $\sqrt{x-3} + \sqrt{x^2-4x+3} = \sqrt{{(x-2)}^3}$ .

(4 points)

Solution (in Hungarian)

B. 4448. The escribed circle drawn to side AC of a triangle ABC touches the lines of sides BC, AC and AB at the points A₁, B₁ and C₁, respectively. Let F denote the midpoint of the line segment A₁B₁. Prove that $\angle$ B₁C₁C= $\angle$ A₁C₁F.

(4 points)

Solution (in Hungarian)

B. 4449. In how many zeros does the number 4^5⁶+6^5⁴ end in decimal notation?

(4 points)

Solution (in Hungarian)

B. 4450. A, B and C are three distinct given points lying on the line e in this order. What is the locus of the points P for which the inscribed circle of triangle ACP touches line e at B?

(4 points)

Solution (in Hungarian)

B. 4451. Consider a convex n-sided polygon that has no four vertices lying on the same circle. Three vertices are selected in every possible way, and the circle passing through the three points is drawn. Such a circle is called thin if the vertices of the polygon (except the vertices of the triangle) all lie outside the circle. The circle is called fat if all vertices of the polygon lie on the closed disc bounded by the circle. Are there more thin circles or more fat circles?

Suggested by S. Róka, Nyíregyháza

(5 points)

Solution (in Hungarian)

Problems sign 'C'

Deadline expired.

C. 1120. Represent those points P(x;y) on the coordinate plane whose coordinates satisfy |y| $\le$ 1-x and |x| $\le$ 3-y.

(5 points)

Solution (in Hungarian)

C. 1121. Prove that if n is a natural number then the value of the sum $\frac{3^{2n}}{112}-\frac{4^{2n}}{63}+\frac{5^{2n}}{144}$ is an integer.

(5 points)

Solution (in Hungarian)

C. 1122. The angle bisector of angle BAC of a triangle ABC intersects side BC at A₁. Let the centres of the circumscribed circles of triangles ABC, ABA₁, ACA₁ be O, O₁, O₂, respectively. Prove that triangle OO₁O₂ is isosceles.

(5 points)

Solution (in Hungarian)

C. 1123. A 4×4×4 cube is built out of sugar cubes. How many different cuboids (that differ in at least one sugar cube) are formed by the sugar cubes?

(5 points)

Solution (in Hungarian)

C. 1124. Solve the following simultaneous equations: x^x+y=y³, y^x+y=x¹².

(5 points)

Solution (in Hungarian)

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