Difference between revisions of "1992 IMO Problems"
(Created page with "Problems of the 1992 IMO. ==Day I== ===Problem 1=== Find all integers <math>a</math>, <math>b</math>, <math>c</math> satisfying <math>1 < a < b < c</math> such that <math...") |
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<cmath>|S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|,</cmath> | <cmath>|S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|,</cmath> | ||
− | where <math>|A|</math> denotes the number of elements in the finite set <math>|A|</math>. | + | where <math>|A|</math> denotes the number of elements in the finite set <math>|A|</math>. |
[[1992 IMO Problems/Problem 5|Solution]] | [[1992 IMO Problems/Problem 5|Solution]] |
Latest revision as of 18:53, 4 July 2024
Problems of the 1992 IMO.
Contents
Day I
Problem 1
Find all integers ,
,
satisfying
such that
is a divisor of
.
Problem 2
Let denote the set of all real numbers. Find all functions
such that
Problem 3
Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of 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.
Day II
Problem 4
In the plane let be a circle,
a line tangent to the circle
, and
a point on
. Find the locus of all points
with the following property: there exists two points
,
on
such that
is the midpoint of
and
is the inscribed circle of triangle
.
Problem 5
Let be a finite set of points in three-dimensional space. Let
,
,
, be the sets consisting of the orthogonal projections of the points of
onto the
-plane,
-plane,
-plane, respectively. Prove that
where denotes the number of elements in the finite set
.
Problem 6
For each positive integer ,
is defined to be the greatest integer such that, for every positive integer
,
can be written as the sum of
positive squares.
(a) Prove that for each
.
(b) Find an integer such that
.
(c) Prove that there are infinitely many integers such that
.
1992 IMO (Problems) • Resources | ||
Preceded by 1991 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1993 IMO |
All IMO Problems and Solutions |