2 Prove that for integer n we have:
n! ≤ n + 1 2
n
(please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)
3 Let {ak}k≥0 be a sequence given by a0 = 0, ak+1= 3 · ak+ 1 for k ∈ N.
Prove that 11 | a155
4 Given a cube in which you can put two massive spheres of radius 1. What’s the smallest possible value of the side - length of the cube? Prove that your answer is the best possible.
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Belgium
Flanders Math Olympiad
1987
3 Find all continuous functions f : R → R such that f (x)3= − x
12 · x
2+ 7x · f (x) + 16 · f (x)2 , ∀x ∈ R.
4 Show that for p > 1 we have lim
n→+∞
1p+ 2p+ ... + (n − 1)p+ np+ (n − 1)p+ ... + 2p+ 1p
n2 = +∞
Find the limit if p = 1.
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Flanders Math Olympiad
1988
1 show that the polynomial x4+ 3x3+ 6x2+ 9x + 12 cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.
2 A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it. The cross is inscribed in a circle with radius 1. What’s its volume?
3 Work base 3. (so each digit is 0,1,2)
A good number of size n is a number in which there are no consecutive 1’s and no consecutive 2’s. How many good 10-digit numbers are there?
4 Be R a positive real number. If R, 1, R +1
2 are triangle sides, call θ the angle between R and R +1
2 (in rad).
Prove 2Rθ is between 1 and π.
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Belgium
Flanders Math Olympiad
1989
1 Show that every subset of 1,2,...,99,100 with 55 elements contains at least 2 numbers with a difference of 9.
2 When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What’s the ratio of the areas of those pentagons?
3 Show that:
α = ±π 12 + k ·
π
2(k ∈ Z) ⇐⇒ |tan α| + |cot α| = 4
4 Let D be the set of positive reals different from 1 and let n be a positive integer. If for f : D → R we have xnf (x) = f (x2), and if f (x) = xn for 0 < x < 1
1989 and for x > 1989, then prove that f (x) = xn for all x ∈ D.
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Flanders Math Olympiad
1990
1 On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 277[/img]Leta and b be two
primes having at least two digits, such that a > b. Show that
240| a4− b4
and show that 240 is the greatest positive integer having this property. 2
3 We form a decimal code of 21 digits. the code may start with 0. Determine the probability that the fragment 0123456789 appears in the code.
4 Let f : R+0 → R+0 be a strictly decreasing function.
(a) Be an a sequence of strictly positive reals so that ∀k ∈ N0 : k · f (ak) ≥ (k + 1) · f (ak+1)
Prove that an is ascending, that lim
k→+∞f (ak) = 0and thatk→+∞lim ak= +∞
(b) Prove that there exist such a sequence (an) in R+0 if you know x→+∞lim f (x) = 0.
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Belgium
Flanders Math Olympiad
1991
1 Show that the number 111...111 with 1991 times the number 1, is not prime.
2 (a) Show that for every n ∈ N there is exactly one x ∈ R+ so that xn+ xn+1 = 1. Call this xn. (b) Find lim
n→+∞xn.
3 Given ∆ABC equilateral, with X ∈ [A, B]. Then we define unique points Y,Z so that Y ∈ [B, C], Z ∈ [A, C], ∆XY Z equilateral.
If Area (∆ABC) = 2 · Area (∆XY Z), find the ratio of AX XB,
BY Y C,
CZ ZA.
4 A word of length n that consists only of the digits 0 and 1, is called a bit-string of length n. (For example, 000 and 01101 are bit-strings of length 3 and 5.) Consider the sequence s(1), s(2), ... of bit-strings of length n > 1 which is obtained as follows : (1) s(1) is the bit- string 00...01, consisting of n − 1 zeros and a 1 ; (2) s(k + 1) is obtained as follows : (a) Remove the digit on the left of s(k). This gives a bit-string t of length n − 1. (b) Examine whether the bit-string t1 (length n, adding a 1 after t) is already in {s(1), s(2), ..., s(k)}. If this is the not case, then s(k + 1) = t1. If this is the case then s(k + 1) = t0.
For example, if n = 3 we get : s(1) = 001 → s(2) = 011 → s(3) = 111 → s(4) = 110 → s(5) = 101 → s(6) = 010 → s(7) = 100 → s(8) = 000 → s(9) = 001 → ...
Suppose N = 2n. Prove that the bit-strings s(1), s(2), ..., s(N ) of length n are all different.
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Flanders Math Olympiad
1992
1 For every positive integer n, determine the biggest positive integer k so that 2k| 3n+ 1 2 It has come to a policeman’s ears that 5 gangsters (all of different height) are meeting, one
of them is the clan leader, he’s the tallest of the 5. He knows the members will leave the building one by one, with a 10-minute break between them, and too bad for him Belgium has not enough policemen to follow all gangsters, so he’s on his own to spot the clanleader, and he can only follow one member.
So he decides to let go the first 2 people, and then follow the first one that is taller than those two. What’s the chance he actually catches the clan leader like this?
3 a conic with apotheme 1 slides (varying height and radius, with r < 1
2) so that the conic’s area is 9 times that of its inscribed sphere. What’s the height of that conic?
4 Let A, B, P positive reals with P ≤ A+B. (a) Choose reals θ1, θ2 with A cos θ1+B cos θ2 = P
and prove that
A sin θ1+ B sin θ2 ≤
p
(A + B − P )(A + B + P ) (b) Prove equality is attained when θ1 = θ2 = arccos
P A + B . (c) Take A =1 2xy, B = 1 2wz and P = 1 4 x 2+ y2− z2− w2 with 0 < x ≤ y ≤ x + z + w, z, w > 0 and z2+ w2 < x2+ y2.
Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts (x, y, z, w), the cyclical one has the greatest area.
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Belgium
Flanders Math Olympiad
1993
1 The 20 pupils in a class each send 10 cards to 10 (different) class members. [note: you cannot send a card to yourself.] (a) Show at least 2 pupils sent each other a card. (b) Now suppose we had n pupils sending m cards each. For which (m, n) is the above true? (That is, find minimal m(n) or maximal n(m))
2 A jeweler covers the diagonal of a unit square with small golden squares in the following way: - the sides of all squares are parallel to the sides of the unit square - for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex) - each midpoint of a square has distance to the vertex of the unit square equal to
1 2,
1 4,
1
8, ... of the diagonal. (so real length: × √
2) - all midpoints are on the diagonal (a) What is the side length of the middle square? (b) What is the total gold-plated area? [img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 281[/img]F ora, b, c > 0 we have:
−1 < a − b a + b 1993 + b − c b + c 1993 + c − a c + a 1993 < 1 3
4 Define the sequence oan as follows: oa0= 1, oan= oan−1· cos
π 2n+1 . Find lim n→+∞oan. http://www.artofproblemsolving.com/
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Flanders Math Olympiad
1994
1 Let a, b, c > 0 the sides of a right triangle. Find all real x for which ax > bx+ cx, with a is the longest side.
2 Determine all integer solutions (a,b,c) with c ≤ 94 for which: (a+√c)2+(b+√c)2 = 60+20√c 3 Two regular tetrahedrons A and B are made with the 8 vertices of a unit cube. (this way is
unique)
What’s the volume of A ∪ B?
4 Let (fi) be a sequence of functions defined by: f1(x) = x, fn(x) =
p
fn−1(x) −
1
4. (n ∈ N, n ≥ 2) (a) Prove that fn(x) ≤ fn−1(x) for all x where both functions are defined. (b) Find for
each n the points of x inside the domain for which fn(x) = x.
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Belgium
Flanders Math Olympiad
1995
1 Four couples play chess together. For the match, they’re paired as follows: (”man Clara” indicates the husband of Clara, etc.)
Bea ⇐⇒ Eddy An ⇐⇒ man Clara F reddy ⇐⇒ woman Guy
Debby ⇐⇒ man An Guy ⇐⇒ woman Eddy Who is Hubert married to?
2 How many values of x ∈ [1, 3] are there, for which x2 has the same decimal part as x? 3 Points A, B, C, D are on a circle with radius R. |AC| = |AB| = 500, while the ratio between
|DC|, |DA|, |DB| is 1, 5, 7. Find R.
4 Given a regular n-gon inscribed in a circle of radius 1, where n > 3. Define G(n) as the average length of the diagonals of this n-gon.
Prove that if n → ∞, G(n) → 4 π.
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Flanders Math Olympiad
1996
1 In triangle ∆ADC we got AD = DC and D = 100◦. In triangle ∆CAB we got CA = AB and A = 20◦.
Prove that AB = BC + CD.
2 Determine the gcd of all numbers of the form p8− 1, with p a prime above 5. 3 Consider the points 1,1
2, 1
3, ... on the real axis. Find the smallest value k ∈ N0 for which all points above can be covered with 5 closed intervals of length 1
k.
4 Consider a real poylnomial p(x) = anxn+ ... + a1x + a0. (a) If deg(p(x)) > 2 prove that
deg(p(x)) = 2 + deg(p(x + 1) + p(x − 1) − 2p(x)). (b) Let p(x) a polynomial for which there are real constants r, s so that for all real x we have
p(x + 1) + p(x − 1) − rp(x) − s = 0 Prove deg(p(x)) ≤ 2. (c) Show, in (b) that s = 0 implies a2 = 0.
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Belgium
Flanders Math Olympiad
1997
1 Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there’s no better solution.
2 In the cartesian plane, consider the curves x2+ y2 = r2 and (xy)2 = 1. Call Fr the convex
polygon with vertices the points of intersection of these 2 curves. (if they exist)
(a) Find the area of the polygon as a function of r. (b) For which values of r do we have a regular polygon?
3 ∆oa1b1is isosceles with ∠a1ob1= 36◦. Construct a2, b2, a3, b3, ... as below, with |oai+1| = |aibi|
and ∠aiobi = 36◦, Call the summed area of the first k triangles Ak. Let S be the area of the
isocseles triangle, drawn in - - -, with top angle 108◦ and |oc| = |od| = |oa1|, going through
the points b2 and a2 as shown on the picture. (yes, cd is parallel to a1b1 there)
Show Ak < S for every positive integer k.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 284[/img]T hirteenbirdsarriveandsitdowninaplane.It0sknownthatf romeach5−
tupleof birds, atleastf ourbirdssitonacircle.DeterminethegreatestM ∈ {1, 2, ..., 13} such that from these 13 birds, at least M birds sit on a circle, but not necessarily M + 1 birds sit on a circle. (prove that your M is optimal)
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Flanders Math Olympiad
1998
1 Prove there exist positive integers a,b,c for which a + b + c = 1998, the gcd is maximized, and 0 < a < b ≤ c < 2a. Find those numbers. Are they unique?
2 Given a cube with edges of length 1, e the midpoint of [bc], and m midpoint of the face cdc1d1,
as on the figure. Find the area of intersection of the cube with the plane through the points a, m, e.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 279[/img]amagical3 × 3 square
is a 3 × 3 matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal.
Determine all magical 3 × 3 square 3
4 A billiard table. (see picture) A white ball is on p1 and a red ball is on p2. The white ball is
shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 280[/img]
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Belgium
Flanders Math Olympiad
1999
1 Determine all 6-digit numbers (abcdef ) so that (abcdef ) = (def )2 where (x1x2...xn) is no
multiplication but an n-digit number.
2 Let [mn] be a diameter of the circle C and [AB] a chord with given length on this circle. [AB] neither coincides nor is perpendicular to [M N ]. Let C, D be the orthogonal projections of A and B on [M N ] and P the midpoint of [AB]. Prove that ∠CP D does not depend on the chord [AB].
3 Determine all f : R → R for which
2 · f (x) − g(x) = f (y) − y and f (x) · g(x) ≥ x + 1.
4 Let a, b, m, n integers greater than 1. If an− 1 and bm+ 1 are both primes, give as much info as possible on a, b, m, n.
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Flanders Math Olympiad
2000
1 An integer consists of 7 different digits, and is a multiple of each of its digits. What digits are in this nubmer?
2 Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles. 3 Let pn be the n-th prime. (p1 = 2) Define the sequence (fj) as follows: - f1 = 1, f2 = 2 -
∀j ≥ 2: if fj = kpn for k < pn then fj+1 = (k + 1)pn - ∀j ≥ 2: if fj = p2n then fj+1 = pn+1
(a) Show that all fi are different (b) from which index onwards are all fi at least 3 digits?
(c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?
4 Solve for x ∈ [0, 2π[:
sin x < cos x < tan x < cot x
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Belgium
Flanders Math Olympiad
2001
1 may be challenge for beginner section, but anyone is able to solve it if you really try. show that for every natural n > 1 we have: (n − 1)2| nn−1− 1
2 Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.
Find the ”?”
3 In a circle we enscribe a regular 2001-gon and inside it a regular 667-gon with shared vertices. Prove that the surface in the 2001-gon but not in the 667-gon is of the form k.sin3
π 2001 .cos3 π 2001 with k a positive integer. Find k.
4 A student concentrates on solving quadratic equations in R. He starts with a first quadratic equation x2+ ax + b = 0 where a and b are both different from 0. If this first equation has solutions p and q with p ≤ q, he forms a second quadratic equation x2 + px + q = 0. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
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Flanders Math Olympiad
2002
1 Is it possible to number the 8 vertices of a cube from 1 to 8 in such a way that the value of the sum on every edge is different?
2 Determine all functions f : R → R so that ∀x : x · f (x 2) − f ( 2 x) = 1 3 show that 1 15 < 1 2 · 3 4· · · 99 100 < 1 10
4 A lamp is situated at point A and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What’s the area of its shadow?
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Belgium
Flanders Math Olympiad
2003
11-12
1 Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc.
Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal.
Who made it?
2 Two circles C1 and C2 intersect at S. The tangent in S to C1 intersects C2 in A different
from S. The tangent in S to C2 intersects C1 in B different from S. Another circle C3 goes
through A, B, S. The tangent in S to C3 intersects C1 in P different from S and C2 in Q
different from S. Prove that the distance P S is equal to the distance QS.
3 A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.
4 Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with r = 2
√
2 goes through 4 points)
Prove that ∀n ∈ N, ∃r so that the circle with midpoint 0,0 and radius r goes through at least n points.
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Flanders Math Olympiad
2003
9-10
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Belgium
Flanders Math Olympiad
2004
11-12
1 Consider a triangle with side lengths 501m, 668m, 835m. How many lines can be drawn with the property that such a line halves both area and perimeter?
2 Two bags contain some numbers, and the total number of numbers is prime.
When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one.
If the total sum of all numbers is 2004, find the number of numbers.
3 A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably)
While the salesmen isn’t watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?
4 Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex T is on the perpendicular line through the center O of the base of the prism (see figure). Let s denote the side of the base, h the height of the cell and θ the angle between the line T O and T V .
(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi. (b) the total surface area of the cell is given by the formula 6sh − 9s
2
2 tan θ+ s23√3 2 sin θ [img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 286[/img]
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Flanders Math Olympiad
2004
9-10
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Belgium
Flanders Math Olympiad
2005
1 For all positive integers n, find the remainder of (7n)!
7n· n! upon division by 7.
2 We can obviously put 100 unit balls in a 10 × 10 × 1 box. How can one put 105 unit balls in? How can we put 106 unit balls in?
3 Prove that 20052 can be written in at least 4 ways as the sum of 2 perfect (non-zero) squares. 4 If n is an integer, then find all values of n for which √n +√n + 2005 is an integer as well.
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Flanders Math Olympiad
2006
1 (a) Solve for θ ∈ R: cos(4θ) = cos(3θ) (b) cos 2π 7 , cos 4π 7 and cos 6π 7
are the roots of an equation of the form ax3+ bx2+ cx + d = 0 where a, b, c, d are integers. Determine a, b, c and d.
2 Let 4ABC be an equilateral triangle and let P be a point on [AB]. Q is the point on BC such that P Q is perpendicular to AB. R is the point on AC such that QR is perpendicular to BC. And S is the point on AB such that RS is perpendicular to AC. Q0 is the point on BC such that P Q0 is perpendicular to BC. R0 is the point on AC such that Q0R0 is perpendicular to AC. And S0 is the point on AB such that R0S0 is perpendicular to AB. Determine |P B| |AB| if S = S0.
3 Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?
4 Find all functions f : R\{0, 1} → R such that f (x) + f 1 1 − x = 1 + 1 x(1 − x). http://www.artofproblemsolving.com/
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