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Problems in Mathematics, September 2010

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 11 October 2010.

A. 512. We have n=\frac{3^k-3}2 coins, each one indistinguishable from the other in terms of size and appearance, but only n-1 of them have the same weight, whilst one of them is a counterfeit, being either lighter or heavier than the rest. Using a balance we have to find the counterfeit coin. Show that by k measurements it is possible to find the counterfeit coin and to determine whether it is heavier or lighter than the rest; moreover, it is possible to arrange the measurements in advance, without knowing the results of the measurements.

(5 points)

Solution, statistics

A. 513. For what primes p does there exist a cubic polynomial f with integer coefficients with the property that p does not divide the leading coefficient of f and f(1),f(2),...,f(p) give pairwise distinct remainders modulo p?

Proposed by Péter Maga

(5 points)

Statistics

A. 514. There are given three circles in the plane, k0, k1 and k2, being externally tangent to each other. The center of k0 is O and one of its diameters is A1A2. Denote by B the point of tangency between k1 and k2, by C1 the point of tangency between k0 and k1 and by C2 the point of tangency between k0 and k2. The line segments A1C2 and A2C1 meet at point D in the interior of the circle k0. Let t1 and t2 be the tangent lines to the circle k0 at A1 and A2, respectively. Prove that if t1 is tangent to k1 and t2 is tangent to k2 then the line segment OB passes through point D.

(5 points)

Statistics


Problems with sign 'B'

Deadline expired on 11 October 2010.

B. 4282. A pool can be filled through four different pipes. If the first and second pipes are operating, it takes 2 hours to fill the pool. Through the second and third pipes it takes 3 hours, and through the third and fourth pipes it takes 4 hours. How long does it take to fill the pool if the first and fourth pipes are operating?

(3 points)

Solution (in Hungarian)

B. 4283. A 23×23 square is divided into smaller squares of dimensions 1×1, 2×2 and 3×3. What is the minimum possible number of 1×1 squares?

(5 points)

Solution (in Hungarian)

B. 4284. Prove that a trapezium circumscribed about a circle has a diagonal that encloses an angle of at most 45o with the bases.

(4 points)

Solution (in Hungarian)

B. 4285. The terms of a sequence are positive integers. The first two terms are 1 and 2. No term of the sequence is equal to the sum of two different terms. Prove that for any natural number k, the number of terms less than k is at most \frac{k}{3} +2.

(3 points)

Solution (in Hungarian)

B. 4286. The leg of an isosceles right angled triangle is 36 units long. Starting at the right-angled vertex on one of the legs, an infinite sequence of regular triangles is drawn. One side of each triangle lies on the leg and the third vertex lies on the hypotenuse. The sides on the leg link together and cover the whole leg. Find the total area of the regular triangles.

(Based on a Kavics Kupa competition problem)

(4 points)

Solution (in Hungarian)

B. 4287. The centres of the escribed circles of triangle ABC are O1, O2 and O3. P is an interior point of the triangle, different from the centre of the inscribed circle. The feet of the perpendiculars drawn from P to the angle bisectors are M1, M2 and M3. Prove that the triangles O1O2O3 and M1M2M3 are similar.

(5 points)

Solution (in Hungarian)

B. 4288. A and B are two opposite vertices of a unit cube. Find the radius of the sphere that touches the faces of the cube that meet in A and the edges that meet in B.

(3 points)

Solution (in Hungarian)

B. 4289. The diagonals of a trapezium A1A2A3A4 are A1A3=e and A2A4=f. Let ri denote the radius of the circumscribed circle of triangle AjAkAl, where {1,2,3,4}={i,j,k,l}. Show that \frac{r_2+r_4}{e}=\frac{r_1+r_3}{f}.

(4 points)

Solution (in Hungarian)

B. 4290. Let a and b denote positive integers. Given that p is a polynomial of integer coefficients that has both a value divisible by a and a value divisible by b at integer points. Prove that there is an integer point where the value of p is also divisible by the least common multiple of a and b.

(5 points)

Solution (in Hungarian)

B. 4291. Prove that the inequality below is true for all positive numbers a, b, c: abbcca\leaabbcc.

(4 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 11 October 2010.

C. 1040. The lawn of a stadium is bounded by two parallel line segments of length 100 metres joined by two 100 metre-long semicircles. By what factor is the area of a circle of perimeter 400 m larger than the area of the lawn?

(5 points)

Solution (in Hungarian)

C. 1041. The digits of a 2010-digit number divisible by nine are added. The digits of the resulting number are also added, and this process is repeated once more. What may be the final result?

(5 points)

Solution (in Hungarian)

C. 1042. Solve the equation x+y=x2-xy+y2, where x and y are integers.

(5 points)

Solution (in Hungarian)

C. 1043. Determine the range of the function given by the following rule of assignment: f(x)=\frac{{(x+a)}^2}{(a-b)(a-c)} + \frac{{(x+b)}^2}{(b-a)(b-c)} +
\frac{{(x+c)}^2}{(c-a)(c-b)}, where a, b and c are different real numbers.

(5 points)

Solution (in Hungarian)

C. 1044. The diagonals of a convex quadrilateral ABCD intersect at point M. Extend diagonal AC beyond A by the length of MC, and extend diagonal BD beyond B by the length of MD to get the points E and F. Prove that EF is parallel to a midline of the quadrilateral.

(5 points)

Solution (in Hungarian)


Problems with sign 'K'

Deadline expired on 11 October 2010.

K. 253. Peter and Paul each chose a number divisible by one hundred. Each of them added one tenth of the chosen number and also one hundredth of it to the original number. The results were 48 507 for Peter and 277 612 for Paul. One boy did the calculation correctly but the other got the last digit wrong. Which answer is correct? What were the original numbers they chose?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 254. A number M is divisible by 14, 15 and 175 but it is not divisible by 28, 45 and 1225. Given that M divides 44 100, find the possible values of M.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 255. What is the minimum number of years needed for the total number of months in them to contain digits of one and zero only?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 256. In a three-digit number, the digit in the ones' place is 3 smaller than the digit in the hundreds' place.

a) Find the largest number that meets the condition.

b) How many such numbers are there altogether?

c) Find all numbers that meet the above condition, for which by subtracting the number obtained by reversing the order of the digits the result is 297.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 257. The surface area of a cuboid is 2010 cm2. If each edge is increased by 1 cm, the surface area will be 2251.52 cm2. Find the sum of the lengths of three different edges of the original cuboid.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 258. x is a positive integer, and the (positive) fraction \frac{1000-x}{1001} can be simplified. How many different values may x have?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


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