Competitions Portal

Problems in Mathematics, May 2008

Please read the rules of the competition.

Problems sign 'A'

Deadline expired.

A. 455. Let H be a set with n elements and let each of the families $\mathcal S$ and $\mathcal T$ consist of p subsets of H such that these 2p subsets are pairwise distinct. Suppose that for every $A\in \mathcal S$ and $B\in \mathcal T$ , the sets A and B have at least one common element. Show that $\frac{p}{2^n}<\frac{3-\sqrt{5}}{2}$ .

Proposed by Ilya Bogdanov, Moscow

(5 points)

A. 456. The point D lies in the triangle ABC such that the circles inscribed in triangles ABD, BCD, and CAD pairwise touch each other. On lines BC, CA, AB, AD, BD, CD, denote the touching points by A₁, B₁, C₁, A₂, B₂, C₂, respectively. Let lines B₁C₂ and B₂C₁ meet at E, and let lines A₁C₂ and A₂C₁ meet at F. Show that the lines AF, BE, and C₁D are concurrent.

(5 points)

Solution (in Hungarian)

A. 457. Prove that, for every odd prime p,

$\sum_{i=1}^{p-1}{2^i i^{p-2}}- \sum_{i=1}^{\frac{p-1}{2}}{i^{p-2}}$

is divisible by p.

(5 points)

Problems sign 'B'

Deadline expired.

B. 4092. Are there four positive integers, such that the greatest common divisor of each pair of them is greater than 1, but the greatest common divisor of each three is 1?

Suggested by P. Szabó

(3 points)

Solution (in Hungarian)

B. 4093. Prove that if n $\ge$ 6 then every triangle can be dissected into n isosceles triangles.

(3 points)

Solution (in Hungarian)

B. 4094. Is it possible to leave one out of the numbers $1, 2, \ldots, 2008$ so that the remaining 2007 numbers can be listed in an order $a_1, a_2, \ldots, a_{2007}$ , in which the differences ${|a_1-a_2|}, {|a_2-a_3|}, \ldots, {|a_{2006}-a_{2007}|}, {|a_{2007}-a_1|}$ are all different?

(4 points)

Solution (in Hungarian)

B. 4095. Given infinitely many cuboids with edges parallel to the axes of the coordinate space, one vertex at the origin and all vertices at points of non-negative integer coordinates. Is it always possible to select two, one of which contains the other?

Javasolta: P. Maga

(5 points)

Solution (in Hungarian)

B. 4096. The points of tangency of the inscribed circle on the sides of a triangle ABC are E, F and G. The distances of an arbitrary point P of the inscribed circle from the sides are a, b and c, and its distances from the lines FG, EG and EF are e, f and g. Prove that abc=efg.

(4 points)

Solution (in Hungarian)

B. 4097. Solve the following equation on the set of integers: $2^{\frac{x-y}{y}} - \frac{3}{2}y = 1$ .

(4 points)

Solution (in Hungarian)

B. 4098. Given a side of a triangle and the two points on it where the lines dividing the opposite angle into three equal parts intersect the side, construct the triangle.

(4 points)

Solution (in Hungarian)

B. 4099. A digit is written in each field of a 10×10 table. Every digit occurs exactly 10 times. Prove that there is a row or column that contains at least 4 different digits.

(Kvant)

(5 points)

Solution (in Hungarian)

B. 4100. Determine the number of sections the plane is divided by the sides of a regular n-gon.

(4 points)

Solution (in Hungarian)

B. 4101. Assume xyz=8. Prove that $\frac{1}{\sqrt{x^2+1}}+\frac{1}{\sqrt{y^2+1}}+\frac{1}{\sqrt{z^2+1}}>1$ .

(5 points)

Solution (in Hungarian)

Problems sign 'C'

Deadline expired.

C. 945. On 1 March, 2008, the Hungarian National Bank withdrew 1 and 2-forint (HUF) coins from circulation. At the cash desk, prices of individual items are not rounded, but the total is rounded to the nearest five. An information leaflet formulates the rule of rounding as follows: Totals ending in 1 or 2 are rounded down to 0; totals ending in 3 or 4 are rounded up to 5; totals ending in 6 or 7 are rounded down to 5; totals ending in 8 or 9 are rounded up to 0. Martin buys two croissants in the little shop on the corner every morning. In a few days, he saves exactly the price of a croissant. Given that a croissant costs more than 10 forints, what may be the price of a croissant so that he can save that amount in the shortest possible time?

(5 points)

Solution (in Hungarian)

C. 946. For what values of the constant c does the equation $x^2- 2\left|x+\frac14\right|+c=0$ have exactly three distinct real roots?

(5 points)

Solution (in Hungarian)

C. 947. A rabbit is sitting at a point A, at a distance of 160 m from a straight railway track. The perpendicular projection of A onto the track is T. A train is approaching T at a speed of 30 m/s. The distance of the front of the train from point T is 300 m initially. The rabbit can run at 15 m/s. Can he cross the track in any direction before the train comes?

(5 points)

Solution (in Hungarian)

C. 948. If one leg of a right-angled triangle is increased by 5 and the other leg is decreased by 5, its area will increase by 5. How will the area of the square drawn over the hypotenuse change?

(5 points)

Solution (in Hungarian)

C. 949. The midpoint of the base AB of an isosceles triangle ABC is F, and its orthocentre is M. Given that the centroid of the triangle lies on the inscribed circle and $FM=\sqrt6$ , what may be the lengths of the sides of the triangle?

Suggested by D. Fülöp, Marcali

(5 points)

Solution (in Hungarian)

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