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Kor MO197

This document contains 8 math problems from the 10th Korean Mathematical Olympiad in 1996. The problems cover a range of topics including geometry (problems 1, 4, 8), number theory (problems 2, 5, 6), combinatorics (problem 7), and algebra (problem 3). The goal is to solve each problem, which test skills like proof by contradiction, algebraic manipulation, counting principles, and properties of circles, triangles and sequences.

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Akachi Nelson
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159 views1 page

Kor MO197

This document contains 8 math problems from the 10th Korean Mathematical Olympiad in 1996. The problems cover a range of topics including geometry (problems 1, 4, 8), number theory (problems 2, 5, 6), combinatorics (problem 7), and algebra (problem 3). The goal is to solve each problem, which test skills like proof by contradiction, algebraic manipulation, counting principles, and properties of circles, triangles and sequences.

Uploaded by

Akachi Nelson
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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10-th Korean Mathematical Olympiad 1996/97

First Round – November 10, 1996

1. Show that √
among any four points in a unit circle, there exist two whose distance
is at most 2.
2. Suppose f : N → N is a function satisfying:
(i) f (n + f (n)) = f (n) for all n;
(ii) For some n0 ∈ N, f (n0 ) = 1.
Prove that f (n) = 1 for all n ∈ N.
√ √
3. Express ∑nk=1 [ k] in terms of n and a = [ n].
4. A circle C touches the edges of an angle XOY , and a circle C1 touches these
edges and passes through the center of C. Let A be the second endpoint of the
diameter of C1 containing the center of C, and let B be the second intersection
point of this diameter with C. Prove that the circle centered at A passing through
B touches the edges of ∠XOY .
5. Find all integers x, y, z satisfying x2 + y2 + z2 = 2xyz.
6. Find the smallest integer k for which there exist two sequences (ai ) and (bi ),
i = 1, 2, . . . , k, such that:
(i) ai , bi ∈ {1, 1996, 19962, . . . } for all i;
(ii) ai 6= bi for all i;
(iii) ai ≤ ai+1 and bi ≤ bi+1 for i = 1, . . . , k − 1;
(iv) ∑ki=1 ai = ∑ki=1 bi .
7. Let An be the set of all real nhumbers of the form
a1 a2 an
1+ √ + √ + ···+ √ , a j ∈ {−1, 1}.
2 ( 2)2 ( 2)n

Find the number of elements of An , and find the sum of all products of two
distinct elements of An .
8. In an acute triangle ABC with AB 6= AC, let V be the intersection of the angle
bisector of A with BC, and let D be the foot of the altitude from A. If the circum-
circle of △AV D meets CA again at E and AB at F, show that the lines AD, BE,CF
concur.

The IMO Compendium Group,


D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com

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