Download Patterns: - Berkeley Math Circle

Berkeley Math Circle
BAMO 2002 practice problems
by Maksim Maydanskiy
Note: The problems below appeared at George Soros Mathematical Olympiads of
Ukraine during 1995 – 1999. I have selected and translated them, occasionally
making slight modifications.
Simply patterns:
1. How many zeroes are in the end of 9 2002  1 ?
2. Is it possible to tile the plane with regular pentagons of side b and rhombs of side
b and acute angle 36°(the tiles can not lie one over other), using:
a) only rhombuses
b) only pentagons
c) both pentagons and rhombuses?
3. Find all ordered triples a, b, c, d of positive integers such that da+db=dc.
4. Solve [x]+[2x]+…+[2002x] = 2002 ([x] denotes the greatest integer not exceeding
x).
5. Find all functions f: R  R such that for all real x and y
f ( x  y )  f ( x) cos y  f ( y ) cos x .
Some geometry:
6. Triangles MAB and MA1B1 are similar and oriented the same way (that is, without
loss of generality, MAB and MA1B1 are both names of the vertices read in clockwise
direction). Prove that circles circumscribed around these triangles and lines AA1 and
BB1 pass through the same point.
7.Circles 1 and  2 intersect in two points – A and B. Chord BC of 1 intersects
 2 in point E (B  E), chord BD of  2 intersects 1 in point F (B  F), so that DF =
CE. Prove that distances from A to segments DF and CE are equal.
1
q
8. Find all triples of positive integers (p, r, q) such that 
1
p qr  1
9. What maximum number of circles of radius 3 4 is it possible to place inside of
the regular hexagon of side 1?(The circles may touch the sides of the hexagon.)
2 x  3x 5
 holds.
10. Prove that for all real x  1x the inequality x
3  5x 7
11. The 2001x2001 board is colored in black and white in a pattern of a chessboard
(black and white squares alternating), so that corner squares are black. A spider sits
on a black square and in one move can move to a square having a common side with
the square it is on. Prove that the spider can always get to the fly, that sits motionless
on some black square, visiting all squares of the board exactly once.
Almost not geometry:
9. In triangle ABC a point of intersections of altitude H lies on the inscribed circle. Is
ABC necessarily isosceles?
And in similar fashion:
10. Does there exist a pentagon each diagonal of which is equal to some side of it?
More geometry (work backward and forward)
11. In a trapezium ABCD (BC||AD, BC<AD, AB=BC=CD) segments CK | AD (K
lies on AD) and KP | AC (P lies on AC) are drawn, so that KP intersects BD at the
point Q. Prove that BQ=QD.
Smaller cases:
12. A positive integer n is called fragile if there exist positive integers a, b, x, and y
x y
such that a  b  n and   1 . Find all fragile numbers.
a b
13. Two players take turns putting matches into a red box and a blue box. First player
during his turn can either put ten matches to the red box, or one match to the blue
box. The second player can either put one match to the red box, or ten matches to the
blue box. At the beginning of the game the boxes are empty. The player after whose
move at least one of the boxes has more than 2001 matches wins. Which player has a
winning strategy?
Specialize:
14. Each point of the disc K is colored in one of m colors (m – given arbitrary
positive integer). Prove that for n  3 exist a color and an infinite set of mutually
congruent n-gons, so that all their vertices are colored with that color.
15. Find all positive integers k for which there exist positive integers m, n such that
m 2  n 2  kmn .


16. Find 2 1  2  3  2  3  4  ...  2001  2002  2003 (here x  stands for the greatest
integer not bigger than x).
17. The triangle ABC and a point M, different from A, B and C are given. Prove that
the area of the triangle whose vertices are the points of the intersection of the medians
of the triangles MAB, MBC, MCA does not depend on the position of point M.
For Extra Die-Hards:
18. Prove that for any integer n>1 exists a integer positive integer m, such that for
some integers k1 , k 2 ,..., k m 4 , different from zero, the we have:
mn  k1  k 2  ...  k m 4  k12001  k 22001  ...  k m2001
4
19. I have cut from a piece of cardboard two equal regular n-gons ( n  3 ), have
covered them with glue and am about to glue them together to make one n-gon. For
which n can I number the vertices of these n-gons with integers 1 through n, so that
for arbitrary way of gluing, vertices in at least one of the pairs glued together have the
same numbers?
Harder:
20. Prove, that for any positive integer m exist integers a and b such that
1 2
.
a  m, b  m and 0  a  b 2 
m2
“Think outside of the board”:
21. Given a table 2002x2002, in each cell of which a real number is written. Two
players, Inna and Neil, take turns choosing rows of the table (at each step any player
can choose any row among those not already chosen). The game ends when all rows
have been chosen. After that, Inna computes the square of the sum of all numbers in
the first column of her 1001 rows, and then she computes the square of the sum of all
numbers in the second column of her 1001 rows etc. Then Inna computes the end
result – the sum of those 2002 squares. Similarly, Neil finds his end result, using his
1001 rows. The one who gets a bigger number wins. If the numbers are equal the
game is considered a draw.
Inna starts the game. Prove that she can play so that she won’t loose.
More geometry:
22. Let ABC be an isosceles triangle with AB=AC. Denote as A1, B1, C1 points of
tangency of the inscribed circle with sides BC, AC, AB respectively. Prove that the
acute angle between lines BB1 and CC1 does not exceed 60 °.
Try smaller sizes, see the pattern:
23. A square of size (n-1)x(n-1) is divided into (n-1)2 one by one squares in a regular
manner. Each of n2 vertices of these squares is to be painted blue or gold. Find the
number of different colorings, for which each unit square has exactly two blue
vertices. (Colorings are called different if they differ in the color of at least one
vertex).
Simpler one:
24. Find the biggest possible number of consecutive positive integers each of which
has exactly 4 positive integer divisors.
25. Prove that in an arbitrary n-gon there exist two sides with lengths a and b, so that
[a / b] = 1 (where [x] denotes the greatest integer not exceeding x).
Yet more geometry:
26. In a rhombus ABCD angle A is equal 60 °. On the rays CD and AD points P and
Q, respectively, are chosen so that AP||CQ. Find the angle PBQ.
27. In the coordinate plane, the circle centered at the origin and radius 2002 is drawn.
In each point of the plain with integer coordinates and inside the circle sits a
gastropod. At some instant of time all gastropods crawl distance 1 up, down, left or
right, staying inside of the circle. Is it necessary that two gastropods will end up at the
same point?
28. Prove that base10 expansion of at least 300 numbers form the set 1, 2, 4, ..., 21000
begins with 1. (Follow-up for die-hards: what is the probability that a number from
the sequence 1, 2, …, 2k, … starts with 1?)
29. (For die-hards again) Points B1 and B2 lie on the median BD of a triangle ABC.
Lines AB2 and AB1, CB2 and CB1 cross BC and BA at the points A2, A1 and C2, C1
respectively. M and K are points of intersection of AA2 and CC1, CC2 and AA1
respectively. Prove that S AKB2  S MA2 A1B1  S CB2 M  S KC2C1B1 .