Difference between revisions of "1990 IMO Problems"
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<cmath> \frac {2^n + 1}{n^2} </cmath> is an integer. | <cmath> \frac {2^n + 1}{n^2} </cmath> is an integer. | ||
− | [[1990 IMO Problems/Problem | + | [[1990 IMO Problems/Problem 3|Solution]] |
==Day II== | ==Day II== | ||
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Given an initial integer <math> n_0 > 1</math>, two players, <math> {\mathcal A}</math> and <math> {\mathcal B}</math>, choose integers <math> n_1</math>, <math> n_2</math>, <math> n_3</math>, <math> \ldots</math> alternately according to the following rules : | Given an initial integer <math> n_0 > 1</math>, two players, <math> {\mathcal A}</math> and <math> {\mathcal B}</math>, choose integers <math> n_1</math>, <math> n_2</math>, <math> n_3</math>, <math> \ldots</math> alternately according to the following rules : | ||
− | I.) Knowing <math> n_{2k}</math>, <math> | + | I.) Knowing <math> n_{2k}</math>, <math>\mathcal{A}</math> chooses any integer <math> n_{2k + 1}</math> such that |
− | + | <cmath>n_{2k} \leq n_{2k + 1} \leq n_{2k}^2.</cmath> | |
− | + | ||
− | II.) Knowing <math> n_{2k + 1}</math>, <math> {\mathcal B}</math> chooses any integer <math> n_{2k + 2}</math> such that | + | II.) Knowing <math> n_{2k + 1}</math>, <math> {\mathcal B}</math> chooses any integer <math> n_{2k + 2}</math> such that <cmath>\dfrac{n_{2k + 1}}{n_{2k + 2}}</cmath> |
− | \ | ||
− | |||
is a prime raised to a positive integer power. | is a prime raised to a positive integer power. | ||
Latest revision as of 17:44, 6 November 2022
Problems of the 1990 IMO.
Contents
Day I
Problem 1
Chords and
of a circle intersect at a point
inside the circle. Let
be an interior point of the segment
. The tangent line at
to the circle through
,
, and
intersects the lines
and
at
and
, respectively. If
find
in terms of
.
Problem 2
Let and consider a set
of
distinct points on a circle. Suppose that exactly
of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly
points from
. Find the smallest value of
so that every such coloring of
points of
is good.
Problem 3
Determine all integers such that
is an integer.
Day II
Problem 4
Let be the set of positive rational numbers. Construct a function
such that
for all
,
in
.
Problem 5
Given an initial integer , two players,
and
, choose integers
,
,
,
alternately according to the following rules :
I.) Knowing ,
chooses any integer
such that
II.) Knowing ,
chooses any integer
such that
is a prime raised to a positive integer power.
Player wins the game by choosing the number 1990; player
wins by choosing the number 1. For which
does :
a.) have a winning strategy?
b.)
have a winning strategy?
c.) Neither player have a winning strategy?
Problem 6
Prove that there exists a convex 1990-gon with the following two properties :
a.) All angles are equal.
b.) The lengths of the 1990 sides are the numbers ,
,
,
,
in some order.
- 1990 IMO
- IMO 1990 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1990 IMO (Problems) • Resources | ||
Preceded by 1989 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1991 IMO |
All IMO Problems and Solutions |