AoPS Community 2001 JBMO ShortLists
JBMO ShortLists 2001
www.artofproblemsolving.com/community/c3731
by WakeUp
2 −10
1 Find the positive integers n that are not divisible by 3 if the number 2n + 2133 is a perfect
cube.
The wording of this problem is perhaps not the best English. As far as I am aware, just solve
2
the diophantine equation x3 = 2n −10 + 2133 where x, n ∈ N and 3 - n.
2 Let Pn (n = 3, 4, 5, 6, 7) be the set of positive integers nk + nl + nm , where k, l, m are positive
integers. Find n such that:
i) In the set Pn there are infinitely many squares.
ii) In the set Pn there are no squares.
3 Find all the three-digit numbers abc such that the 6003-digit number abcabc . . . abc is divisible
by 91.
4 The discriminant of the equation x2 − ax + b = 0 is the square of a rational number and a and
b are integers. Prove that the roots of the equation are integers.
5 Let xk = k(k+1)
2 for all integers k ≥ 1. Prove that for any integer n ≥ 10, between the numbers
A = x1 + x2 + . . . + xn−1 and B = A + xn there is at least one square.
6 Find all integers x and y such that x3 ± y 3 = 2001p, where p is prime.
7 Prove that there are are no positive integers x and y such that x5 + y 5 + 1 = (x + 2)5 + (y − 3)5 .
The restriction x, y are positive isn’t necessary.
8 Prove that no three points with integer coordinates can be the vertices of an equilateral trian-
gle.
9 Consider a convex quadrilateral ABCD with AB = CD and ∠BAC = 30◦ . If ∠ADC = 150◦ ,
prove that ∠BCA = ∠ACD.
10 A triangle ABC is inscribed in the circle C(O, R). Let α < 1 be the ratio of the radii of the circles
tangent to C, and both of the rays (AB and (AC. The numbers β < 1 and γ < 1 are defined
analogously. Prove that α + β + γ = 1.
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AoPS Community 2001 JBMO ShortLists
11 Consider a triangle ABC with AB = AC, and D the foot of the altitude from the vertex A. The
point E lies on the side AB such that ∠ACE = ∠ECB = 18◦ .
If AD = 3, find the length of the segment CE.
12 Consider the triangle ABC with ∠A = 90◦ and ∠B 6= ∠C. A circle C(O, R) passes through B
and C and intersects the sides AB and AC at D and E, respectively. Let S be the foot of the
perpendicular from A to BC and let K be the intersection point of AS with the segment DE.
If M is the midpoint of BC, prove that AKOM is a parallelogram.
13 At a conference there are n mathematicians. Each of them knows exactly k fellow mathemati-
cians. Find the smallest value of k such that there are at least three mathematicians that are
acquainted each with the other two.
Rewording of the last line for clarification:
Find the smallest value of k such that there (always) exists 3 mathematicians X, Y, Z such
that X and Y know each other, X and Z know each other and Y and Z know each other.
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