MathDB

2008 All-Russian Olympiad

Part of All-Russian Olympiad

Subcontests

(8)
1
1
5
3

Cells of a chessboard with distance 50

The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to 100 100. How many cells are there that are on the distance 50 50 from each of the three cells?

Find all triplets satisfying to inequalities.

Determine all triplets of real numbers x,y,z x,y,z satisfying 1\plus{}x^4\leq 2(y\minus{}z)^2,  1\plus{}y^4\leq 2(x\minus{}z)^2,  1\plus{}z^4\leq 2(x\minus{}y)^2.

integral roots

The numbers from 51 51 to 150 150 are arranged in a 10×10 10\times 10 array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers a a and b b, at least one of the equations x^2 \minus{} ax \plus{} b \equal{} 0 and x^2 \minus{} bx \plus{} a \equal{} 0 has two integral roots?
2
3

Cubic polynomial has root in segment [0,2]

Numbers a,b,c a,b,c are such that the equation x^3 \plus{} ax^2 \plus{} bx \plus{} c has three real roots.Prove that if \minus{} 2\leq a \plus{} b \plus{} c\leq 0,then at least one of these roots belongs to the segment [0,2] [0,2]

arrangement of good cells

The columns of an n×n n\times n board are labeled 1 1 to n n. The numbers 1,2,...,n 1,2,...,n are arranged in the board so that the numbers in each row and column are pairwise different. We call a cell "good" if the number in it is greater than the label of its column. For which n n is there an arrangement in which each row contains equally many good cells?

Find the smallest possible value

Petya and Vasya are given equal sets of N N weights, in which the masses of any two weights are in ratio at most 1.25 1.25. Petya succeeded to divide his set into 10 10 groups of equal masses, while Vasya succeeded to divide his set into 11 11 groups of equal masses. Find the smallest possible N N.
4
3

Largest possible number of scientist

There are several scientists collaborating in Niichavo. During an 8 8-hour working day, the scientists went to cafeteria, possibly several times.It is known that for every two scientist, the total time in which exactly one of them was in cafeteria is at least x x hours (x>4 x>4). What is the largest possible number of scientist that could work in Niichavo that day,in terms of x x?

upper bound of 2008th term

The sequences (an),(bn) (a_n),(b_n) are defined by a_1\equal{}1,b_1\equal{}2 and a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n},   b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.
Show that a2008<5 a_{2008} < 5.

Tetrahedron

Each face of a tetrahedron can be placed in a circle of radius 1 1. Show that the tetrahedron can be placed in a sphere of radius 322 \frac{3}{2\sqrt2}.
8
2
7
3

2008 appears on blackboard

A natural number is written on the blackboard. Whenever number x x is written, one can write any of the numbers 2x \plus{} 1 and \frac {x}{x \plus{} 2}. At some moment the number 2008 2008 appears on the blackboard. Show that it was there from the very beginning.

product of sum becoming exponents

For which integers n>1 n>1 do there exist natural numbers b1,b2,…,bn b_1,b_2,\ldots,b_n not all equal such that the number (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k) is a power of an integer for each natural number k k? (The exponents may depend on k k, but must be greater than 1 1)

harrrrrd quadrilateral property

In convex quadrilateral ABCD ABCD, the rays BA,CD BA,CD meet at P P, and the rays BC,AD BC,AD meet at Q Q. H H is the projection of D D on PQ PQ. Prove that there is a circle inscribed in ABCD ABCD if and only if the incircles of triangles ADP,CDQ ADP,CDQ are visible from H H under the same angle.
3
3

Collinear

In a scalene triangle ABC,H ABC, H and M M are the orthocenter an centroid respectively. Consider the triangle formed by the lines through A,B A,B and C C perpendicular to AM,BM AM,BM and CM CM respectively. Prove that the centroid of this triangle lies on the line MH MH.

Segment equal

A circle ω \omega with center O O is tangent to the rays of an angle BAC BAC at B B and C C. Point Q Q is taken inside the angle BAC BAC. Assume that point P P on the segment AQ AQ is such that AQ⊥OP AQ\perp OP. The line OP OP intersects the circumcircles ω1 \omega_{1} and ω2 \omega_{2} of triangles BPQ BPQ and CPQ CPQ again at points M M and N N. Prove that OM \equal{} ON.

write in form a^p+b^p

Given a finite set P P of prime numbers, prove that there exists a positive integer x x such that it can be written in the form a^p \plus{} b^p (a,b a,b are positive integers), for each p∈P p\in P, and cannot be written in that form for each p p not in P P.
6
3

Parallel

In a scalene triangle ABC ABC the altitudes AA1 AA_{1} and CC1 CC_{1} intersect at H,O H, O is the circumcenter, and B0 B_{0} the midpoint of side AC AC. The line BO BO intersects side AC AC at P P, while the lines BH BH and A1C1 A_{1}C_{1} meet at Q Q. Prove that the lines HB0 HB_{0} and PQ PQ are parallel.

9th grade 2008 Russian Mathematical Olympiad 6th question

The incircle of a triangle ABCABC touches the side ABAB and ACAC at respectively at XX and YY. Let KK be the midpoint of the arc AB^\widehat{AB} on the circumcircle of ABCABC. Assume that XYXY bisects the segment AKAK. What are the possible measures of angle BACBAC?

Magician

A magician should determine the area of a hidden convex 2008 2008-gon A1A2⋯A2008 A_{1}A_{2}\cdots A_{2008}. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in 2006 2006 steps.