Difference between revisions of "1994 IMO Problems"
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===Problem 5=== | ===Problem 5=== | ||
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+ | Let <math>S</math> be the set of real numbers strictly greater than <math>-1</math>. Find all functions <math>f:S \to S</math> satisfying the two conditions: | ||
+ | |||
+ | 1. <math>f(x+f(y)+xf(y)) = y+f(x)+yf(x)</math> for all <math>x</math> and <math>y</math> in <math>S</math>; | ||
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+ | 2. <math>\frac{f(x)}{x}</math> is strictly increasing on each of the intervals <math>-1<x<0</math> and <math>0<x</math>. | ||
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+ | [[1994 IMO Problems/Problem 5 | Solution]] | ||
===Problem 6=== | ===Problem 6=== | ||
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+ | Show that there exists a set <math>A</math> of positive integers with the following property: For | ||
+ | any infinite set <math>S</math> of primes there exist two positive integers <math>m \in A</math> and <math>n \not\in A</math> each of which is a product of <math>k</math> distinct elements of <math>S</math> for some <math>k \ge 2</math>. | ||
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+ | [[1994 IMO Problems/Problem 6 | Solution]] | ||
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* [[1994 IMO]] | * [[1994 IMO]] |
Latest revision as of 13:01, 19 April 2024
Problems of the 1994 IMO.
Contents
[hide]Day I
Problem 1
Let and be two positive integers. Let , , , be different numbers from the set such that for any two indices and with and , there exists an index such that . Show that .
Problem 2
Let be an isosceles triangle with . is the midpoint of and is the point on the line such that is perpendicular to . is an arbitrary point on different from and . lies on the line and lies on the line such that are distinct and collinear. Prove that is perpendicular to if and only if .
Problem 3
For any positive integer , let be the number of elements in the set whose base 2 representation has precisely three s.
- (a) Prove that, for each positive integer , there exists at least one positive integer such that .
- (b) Determine all positive integers for which there exists exactly one with .
Day II
Problem 4
Find all ordered pairs where and are positive integers such that is an integer.
Problem 5
Let be the set of real numbers strictly greater than . Find all functions satisfying the two conditions:
1. for all and in ;
2. is strictly increasing on each of the intervals and .
Problem 6
Show that there exists a set of positive integers with the following property: For any infinite set of primes there exist two positive integers and each of which is a product of distinct elements of for some .
- 1994 IMO
- IMO 1994 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1994 IMO (Problems) • Resources | ||
Preceded by 1993 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1995 IMO |
All IMO Problems and Solutions |