Difference between revisions of "1990 IMO Problems"
(Created page with "Problems of the 1990 IMO. ==Day I== ===Problem 1=== Chords <math> AB</math> and <math> CD</math> of a circle intersect at a point <math> E</math> inside the circle. Let <...") |
(→Problem 5) |
||
Line 31: | Line 31: | ||
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. | ||
Revision as of 00:20, 10 August 2021
Problems of the 1990 IMO.
Contents
[hide]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 |