NOTE: If you have any bright ideas please mail me
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1.Does there exist a non-trivial continuous function f:R -> R, which satisfies f(x) + f(2x) + f(3x)
= 0 for all x?
© 1997 andysc@my-dejanews.com
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2. Given k a fixed non-negative integer, and P(2x) = 2^{k - 1}(P(x) + P(x + 1/2)). Prove that
P(3x) = 3^{k - 1}(P(x) + P(x + 1/3) + P(x + 2/3)).
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3.Let a, b, c be integers and p an odd prime. Prove that if f(x) = ax^2 + bx + c is a perfect square
for 2p - 1 consecutive integer values of x, then p divides b^2 - 4ac.
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4. Prove that a closed semi-disc of diameter 1 can cover any worm of length 1 in the plane.
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5.The polynomial f(x) has real coefficients, and satisfies f(x) > 0 for x >= 0. Prove that there
exists an integer m such that (1 + x)^m f(x) has non-negative coefficients.
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6.The hexagon ABCDEF has sides all of length 1. Prove that not all the diagonals AD, BE, and
CF can have length greater than 2.
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7.The sequence {a_n} is decreasing, and is such that a_n >= 1/n for infinitely many positive
integers n. Show that \sum_1^inf a_i diverges.
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8.Consider a set of rectangles, where the n^th rectangle has dimensions 1/n x 1/(n+1), n = 1, 2, 3,
... . The sum of their areas is 1. Can they tile the unit square?
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The above are problems originally posted by Naoki Sato
Naoki’s mail address
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Spring Mathematics Olympiad Problems
Organized by
the Palace of Youth Creativity
St. Petersburg, Russia
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9.One bacterium divides into two new ones. Then every second bacterium also splits in two, and
soon there are 1995 bacteria.
Prove that one of them lives for at least 995 seconds without any division.
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10.Divide a square into 7 polygons such that any two like polygons will have no common edge,
and any two that are not the same shape will have a common edge. (For any polygon there must
exist another with the same shape.)
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11. Prove that there are infinitely many integers that are not equal to the sum of two squares
of integers, but that are equal to the sum of three cubes of integers.
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12. Is it possible to put the numbers from 1 to 64 in the squares of a chess board in such a
way that the sum of the numbers in any three squares arranged in the following pattern will be
odd?
SQ1
SQ2..........SQ3
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13. Let S be a square, Q be the perimeter of the square, and P be the perimeter of a
quadrilateral T inscribed within S such that each of its vertices lies on a different edge of S.
What is the smallest possible ratio of P to Q?
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14. X and Y are two integers satisfying 1 < X <= 100 and 1 < Y <= 100.
Mr. S knows their sum (X+Y). Mr. P knows their product (X*Y).
You overhear the following conversation:
Mr. S: I don't know X and Y.
Mr. P: I knew you didn't. Neither do I.
Mr. S: Now I know X and Y.
Mr. P: Now I do too.
14.1. Is this enough information to figure out what X and Y are?
14.2. If so, what are they? If not, what are the possibilities?
(Assume that Mr. S and Mr. P are both logical and truthful. Also assume that Mr. S knows that
Mr. P knows the product of X and Y and vice versa, and that both know the possible range of X
and Y.)
Warning: This problem may take quite a while to solve. Also, I make no guarantee that an
"elegant" solution exists.
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15. In a 4 x 4 chessboard, all squares are coloured white or black. Given an initial colouring
of the board, we are allowed to recolour it by changing the colour of all squares in any 3 x 3 or 2
x 2 subboard. Is it possible to get every possible recoloring of the board from the all white
colouring by spplying some number of these “subboard colourings”?
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16. Suppose we have 13 integers with the property that if you remove any one of them, then
the remaining 12 can be partitioned into two sets of 6 with equal sums . Prove that all 13 integers
are equal.
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17. UNDER CONSTRUCTION, anyone know how to write equations properly in HTML?
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18. UNDER CONSTRUCTION, anyone know how to get an integral sign in HTML?
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19. Which MxN boards can be tiled by 1x4 tiles?
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20. Estimate the difference between phi^10 and the nearest integer, where phi is the golden ratio.
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21. Count the number of odd entries in the 100th row of Pascal's triangle.
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22. Prove or disprove: There exists a function f from R to R such that
i) f sends rationals to rationals and irrationals to irrationals.
ii) f is not linear on any interval.
iii) f is differentiable.
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23. Can you find 6 lattice points (integral coordinates) which are the vertices of a regular
hexagon?
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24. If g(x)=g(x+c) for some x, call c a chord of g. Which real numbers are chords of all
continuous functions f with domain [0,1] such that f(0)=f(1)?
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25. Which polygons can be geometrically dissected and reassembled into a square?
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26. Let f be any monic degree n polynomial in x. What is the limit as x goes to infinity of (the
nth root of f(x))-x?
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27. Find a monotone function f:R+-->R+ such that f grows faster than any polynomial, yet
f(f(...n times...f(x))...) grows more slowly than c^x for any c>1 and any positive integer n
.
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28. One can easily pack 2n circles of unit diameter in a 2xn rectangle. Show that there is some n
such that 2n+1 circles may be packed into a 2xn rectangle.
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29. Can one express any positive rational number as the sum of the reciprocals of finitely many
distinct positive integers?
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30. Determine the intersection of the unit hypercube [0,1]^4 with the hyperplane x+y+z+w=2.
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31. Does there exist a continuous function on [0,1] which attains each of its values a finite, even
number of times?
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32. Prove that there exists a unique function f from the set R^+ to R^+ such
that f(f(x)) = 6x - f(x) and f(x) > 0 for all x > 0.
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33. Let w (omega) be a primitive p^th root of unity, where p is a prime, i.e.
w^p = 1, w != 1. Compute (1 - w)(1 - w^2)...(1 - w^(p-1)).
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34. Let a, b, c, and d be distinct complex numbers. Show that
a^4/(a-b)(a-c)(a-d) + b^4/(b-a)(b-c)(b-d) + c^4/(c-a)(c-b)(c-d) + d^4/(d-a)(d-b)(d-c)
= a + b + c + d.
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35. Let F_n denote the n^th Fibonacci number. Calculate 1/F_1 + 1/F_2 + 1/F_4 +
1/F_8 + ... + 1/F_(2^k) + ... .
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36. (Binomial coefficients) Let S_n = 1/C(n,0) + 1/C(n,1) + ... + 1/C(n,n). Find lim_(n -> oo)
S_n.
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37. Let P_n(x) be the sequence defined as follows: P_0(x) = 2, and P_1(x) = x,
and P_(n+1)(x) = xP_n(x) - P_(n-1)(x). Show that all the roots of P_n(x)
are real.
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38. The real numbers a, b, c are such that there is a unique square of side s
whose vertices lie on the graph of y = x^3 + ax^2 + bx + c. Prove that
s^4 = 72.
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39. Show that the points A(a,a^2), B(b,b^2), C(c,c^2), and D(d,d^2) in the
plane, lie on a circle iff a + b + c + d = 0.
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40. Show that there is no set of consecutive positive integers, such that the
sum of their reciprocals is also an integer.
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41. The real valued function f satisfies f(tan 2x) = tan^4 x + cot^4 x. Prove
that f(sin x) + f(cos x) >= 196 for all real x.
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42. A random number is chosen in [0,1] and a running sum is kept. What is the
expected number of times a number is chosen before the sum exceeds 1?
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43. Let x be a positive real. Show that
x+1 = sqrt(1 + x sqrt(1 + (x+1)sqrt(1 + (x+2) sqrt(1 + ...)))).
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44. Is there a convex polygon such that for any vertex v, there are 4 other vertices which are of
equal distance from v?
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45. Given a connected collection of n unit circles in the plane, are there at least n-1 points of
intersection?
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46. For any polynomial P(x) = a + bx + cx^2 + ..... +mx^k (co-efficients not necessarily in
alphabetic order) with integer co-efficients which are odd is denoted by w(P). For i = 0,1,...., let
Q_i (x) = (1 + x)^i. Prove that if i_1, i_2, i_3, i_4,....,i_n are integers such that 0=< i_1< i_2<
i_3<.......
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47. Each of the numbers x_1, x_2, x_3,......, x_n equals 1 or -1 and
x_1*x_2*x_3*x_4 + x_2*x_3*x_4*x_5 + .... +x_n-1*x_n*x_1*x_2 + x_n*x_1*x_2*x_3 = 0
prove n is divisible by 4.
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48. A convex quadrilateral is inscribed in a circle of unit radius 1. Prove that the (positive)
difference u between its perimeter and the sum of the lengths of its diagonals satisfies 0
STILL UNDER CONSTRUCTION!!
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