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Difference between revisions of "1997 IMO Problems"

(Created page with "Find all pairs <math>(a,b)</math> of integers <math>a,b \ge 1</math> that satisfy the equation <math>a^{b^{2}}=b^{a}</math>")
 
 
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Problems of the 1997 [[IMO]].
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==Day I==
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===Problem 1===
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In the plane the points with integer coordinates are the vertices of unit squares.  The squares are colored alternatively black and white (as on a chessboard).
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For any pair of positive integers <math>m</math> and <math>n</math>, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths <math>m</math> and <math>n</math>, lie along edges of the squares.
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Let <math>S_{1}</math> be the total area of the black part of the triangle and <math>S_{2}</math> be the total area of the white part.
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Let <math>f(m,n)=|S_{1}-S_{2}|</math>
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(a) Calculate <math>f(m,n)</math> for all positive integers <math>m</math> and <math>n</math> which are either both even or both odd.
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(b) Prove that <math>f(m,n) \le \frac{1}{2} max\left\{ m,n \right\}</math> for all <math>m</math> and <math>n</math>.
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(c) Show that there is no constant <math>C</math> such that <math>f(m,n)<C</math> for all <math>m</math> and <math>n</math>.
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[[1997 IMO Problems/Problem 1|Solution]]
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===Problem 2===
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The angle at <math>A</math> is the smallest angle of triangle <math>ABC</math>. The points <math>B</math> and <math>C</math> divide the circumcircle of the triangle into two arcs.  Let <math>U</math> be an interior point of the arc between <math>B</math> and <math>C</math> which does not contain <math>A</math>.  The perpendicular bisectors of <math>AB</math> and <math>AC</math> meet the line <math>AU</math> and <math>V</math> and <math>W</math>, respectively.  The lines <math>BV</math> and <math>CW</math> meet at <math>T</math>. Show that.
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[[1997 IMO Problems/Problem 2|Solution]]
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===Problem 3===
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Let <math>x_{1}</math>, <math>x_{2}</math>,...,<math>x_{n}</math> be real numbers satisfying the conditions
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<math>|x_{1}+x_{2}+...+x_{n}|=1</math>
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and
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<math>|x_{i}| \le \frac{n+1}{2}</math>, for <math>i=1,2,...,n</math>
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Show that there exists a permutation <math>y_{1}</math>, <math>y_{2}</math>,...,<math>y_{n}</math> of <math>x_{1}</math>, <math>x_{2}</math>,...,<math>x_{n}</math> such that
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<math>|y_{1}+2y_{2}+...+ny_{n}|\le \frac{n+1}{2}</math>
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[[1997 IMO Problems/Problem 3|Solution]]
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==Day II==
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===Problem 4===
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An <math>n \times n</math> matrix whose entries come from the set <math>S={1,2,...,2n-1}</math> is called a <math>\textit{silver}</math> matrix if, for each <math>i=1,2,...,n</math>, the <math>i</math>th row and the <math>i</math>th column together contain all elements of <math>S</math>.  Show that
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(a) there is no <math>\textit{silver}</math> matrix for <math>n=1997</math>;
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(b) <math>\textit{silver}</math> matrices exist for infinitely many values of <math>n</math>.
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[[1997 IMO Problems/Problem 4|Solution]]
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===Problem 5===
 
Find all pairs <math>(a,b)</math> of integers <math>a,b \ge 1</math> that satisfy the equation
 
Find all pairs <math>(a,b)</math> of integers <math>a,b \ge 1</math> that satisfy the equation
  
 
<math>a^{b^{2}}=b^{a}</math>
 
<math>a^{b^{2}}=b^{a}</math>
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[[1997 IMO Problems/Problem 5|Solution]]
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===Problem 6===
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For each positive integer <math>n</math>, let <math>f(n)</math> denote the number of ways of representing <math>n</math> as a sum of powers of <math>2</math> with nonnegative integer exponents.  Representations which differ only in the ordering of their summands are considered to be the same.  For instance, <math>f(4)=4</math>, because the number 4 can be represented in the following four ways:
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<math>4;2+2;2+1+1;1+1+1+1</math>
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Prove that, for any integer <math>n \ge 3</math>,
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<math>2^{n^{2}/4}<f(2^{n})<2^{n^{2}/2}</math>.
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[[1997 IMO Problems/Problem 6|Solution]]
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==See Also==
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* [[1997 IMO]]
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competition resources]]
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{{IMO box|year=1997|before=[[1996 IMO]]|after=[[1998 IMO]]}}

Latest revision as of 20:47, 4 July 2024

Problems of the 1997 IMO.

Day I

Problem 1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternatively black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares.

Let $S_{1}$ be the total area of the black part of the triangle and $S_{2}$ be the total area of the white part.

Let $f(m,n)=|S_{1}-S_{2}|$

(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

(b) Prove that $f(m,n) \le \frac{1}{2} max\left\{ m,n \right\}$ for all $m$ and $n$.

(c) Show that there is no constant $C$ such that $f(m,n)<C$ for all $m$ and $n$.

Solution

Problem 2

The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ and $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that.

Solution

Problem 3

Let $x_{1}$, $x_{2}$,...,$x_{n}$ be real numbers satisfying the conditions

$|x_{1}+x_{2}+...+x_{n}|=1$

and

$|x_{i}| \le \frac{n+1}{2}$, for $i=1,2,...,n$

Show that there exists a permutation $y_{1}$, $y_{2}$,...,$y_{n}$ of $x_{1}$, $x_{2}$,...,$x_{n}$ such that

$|y_{1}+2y_{2}+...+ny_{n}|\le \frac{n+1}{2}$

Solution

Day II

Problem 4

An $n \times n$ matrix whose entries come from the set $S={1,2,...,2n-1}$ is called a $\textit{silver}$ matrix if, for each $i=1,2,...,n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

(a) there is no $\textit{silver}$ matrix for $n=1997$;

(b) $\textit{silver}$ matrices exist for infinitely many values of $n$.

Solution

Problem 5

Find all pairs $(a,b)$ of integers $a,b \ge 1$ that satisfy the equation

$a^{b^{2}}=b^{a}$

Solution

Problem 6

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number 4 can be represented in the following four ways:

$4;2+2;2+1+1;1+1+1+1$

Prove that, for any integer $n \ge 3$,

$2^{n^{2}/4}<f(2^{n})<2^{n^{2}/2}$.

Solution

See Also

1997 IMO (Problems) • Resources
Preceded by
1996 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1998 IMO
All IMO Problems and Solutions
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