Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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...")
 
m
 
Line 30: Line 30:
 
<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>. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane)
+
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 17:53, 4 July 2024

Problems of the 1992 IMO.

Day I

Problem 1

Find all integers $a$, $b$, $c$ satisfying $1 < a < b < c$ such that $(a - 1)(b -1)(c - 1)$ is a divisor of $abc - 1$.

Solution

Problem 2

Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that

\[f\left( x^{2}+f(y) \right)= y+(f(x))^{2} \hspace{0.5cm} \forall x,y \in \mathbb{R}\]

Solution

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 $n$ 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.

Solution

Day II

Problem 4

In the plane let $C$ be a circle, $l$ a line tangent to the circle $C$, and $M$ a point on $l$. Find the locus of all points $P$ with the following property: there exists two points $Q$, $R$ on $l$ such that $M$ is the midpoint of $QR$ and $C$ is the inscribed circle of triangle $PQR$.

Solution

Problem 5

Let $S$ be a finite set of points in three-dimensional space. Let $S_{x}$,$S_{y}$,$S_{z}$, be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that

\[|S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|,\]

where $|A|$ denotes the number of elements in the finite set $|A|$.

Solution

Problem 6

For each positive integer $n$, $S(n)$ is defined to be the greatest integer such that, for every positive integer $k \le S(n)$, $n^{2}$ can be written as the sum of $k$ positive squares.

(a) Prove that $S(n) \le n^{2}-14$ for each $n \ge 4$.

(b) Find an integer $n$ such that $S(n)=n^{2}-14$.

(c) Prove that there are infinitely many integers $n$ such that $S(n)=n^{2}-14$.

Solution

1992 IMO (Problems) • Resources
Preceded by
1991 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1993 IMO
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1992_IMO_Problems&oldid=221659"