Problems Shortlisted to the 1992 IMO Jury
1. Let m be a positive integer. If there exist two coprime integers a, b such that a|m+ b2
and b|m + a2 then show that we can also find two such coprime integers with the
additional restriction that a, b > 0 and a+ b ≤ m+ 1
2. Let a and b be positive reals. Show that there is a unique function f : R −→ R such
that f(f(x)) = b(a+ b)f(x)− af(x) for all x.
3. The quadrilateral ABCD has perpendicular diagonals. Construct the squares ABEF ,
BCGH, CDIJ and DAKL in the exterior of the quadrangle. Let CL ∩DF = {P},
DF ∩ AH = {Q}, AH ∩ BJ = {R}, BJ ∩ CL = {S}. The lines AI and BK meet
at P ′, the lines BK and CE meet at Q′, the lines CE and DG meet at R′, and the
lines DG and AI meet at S ′. Show that the quadrilaterals PQRS and P ′Q′R′S ′ are
congruent.
4. A convex quadrilateral has equal diagonals. Equilateral triangles are constructed on
the outside of each side. The centers of the triangles on opposite sides are joined. Show
that the two joining lines are perpendicular.
5. Two circles touch externally at I. The two circles lie inside a large circle and both
touch it. The chord BC of the large circle touches both smaller circles (not at I). The
common tangent to the two smaller circles at I meets the large circle at A, where A
and I are on the same side of the chord BC. Show that I is the incenter of ∆ABC.
6. Show that there is an equiangular 1992-gon (all angles equal) with sides 12, 22, . . . , 19922
in some order.
7. Let P (x) be a polynomial with rational coefficients such that k3 − k = 331992 =
P 3(k) − P (k) for some real number k. Let Pn(x) = P (P (. . . P (x) . . .)) (iterated n
times). Show that P 3n(k)− Pn(k) = 331992.
8. Let BD and CE be the angle bisectors of the triangle ABC. ∠BDE = 24◦ and
∠CED = 18◦. Find angles A,B,C.
9. The polynomials f(x), g(x) and a(x, y) have real coefficients. They satisfy f(x) −
f(y) = a(x, y)(g(x)− g(y)) for all x, y. Show that there is a polynomial h(x) such that
f(x) = h(g(x)) for all x.
10. For each positive integer n, let d(n) be the largest odd divisor of n and define f(n) =
n
2
+ n
d(n)
for n even and f(n) = 2(n+1)/2 for n odd. Define the sequence a1, a2, . . . by
a1 = 1, an+1 = f(an). Show that 1992 occurs in the sequence and find the first time it
occurs. Does it occur more than once?
11. Does there exist a set of 1992 positive integers such that each subset has a sum which
is a square, cube or higher power?
1
12. Show that 5
125−1
525−1 is composite.
13. Let b(n) be the number of 1’s in the binary representation of a positive integer n. Show
that b(n2) ≤ b(n)(b(n)+1)
2
with equality for infinitely many positive integers n. Show that
there is a sequence of positive integers a1 < a2 < a3 < . . . such that
b(a2n)
b(an)
tends to zero.
14. Let x1 be a real number such that 0 < x1 < 1. Define xn+1 =
1
xn
−b 1
xn
c for xn non-zero
and 0 for xn zero. Show that x1+ x2+ . . .+ xn <
F1
F2
+ F2
F3
+ . . .+ Fn
Fn+1
, where Fn is the
Fibonacci sequence (F1 = F2 = 1, Fn+2 = Fn+1 + Fn).
15. Find all integers a, b, c so that 1 < a < b < c and (a− 1)(b− 1)(c− 1)|abc− 1.
16. Find all functions f : R −→ R so that f(x2 + f(y)) = y + f 2(x) for all x, y.
17. Consider 9 points in space, no 4 coplanar. Each pair of points is joined by a line
segment which is colored either blue or red or left uncolored. Find the smallest value
of n such that whenever exactly n edges are colored, the set of colored edges necessarily
contains a triangle all of whose edges have the same color.
18. Let L be a tangent to the circle C and let M be a point on L. Find the locus of all
points P such that there exist points Q and R on L equidistant from M with C the
incircle of the triangle PQR.
19. Let S be a finite set of points in three-dimensional space. Let Sx,Sy,Sz be the sets
consisting of the orthogonal projections of the points of S onto the yz-plane, zx-plane,
xy-plane respectively. Prove that
|S|2 ≤ |Sx||Sy||Sz|
where |A| denotes the number of points in the set A.
20. For each positive integer n, S(n) is defined as the greatest integer such that for every
positive integer k ≤ S(n), n2 can be written as the sum of k positive squares.
a. Prove that S(n) ≤ n2 − 14 for each n ≥ 4.
b. Find an integer n such that S(n) = n2 − 14.
c. Prove that there are infinitely many integers n such that S(n) = n2 − 14.
2
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