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Difference between revisions of "2000 IMO Problems"

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== Day 1 ==
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== Problem 1 ==
 
== Problem 1 ==
 
Two circles <math>G_1</math> and <math>G_2</math> intersect at two points <math>M</math> and <math>N</math>. Let <math>AB</math> be the line tangent to these circles at <math>A</math> and <math>B</math>, respectively, so that <math>M</math> lies closer to <math>AB</math> than <math>N</math>. Let <math>CD</math> be the line parallel to <math>AB</math> and passing through the point <math>M</math>, with <math>C</math> on <math>G_1</math> and <math>D</math> on <math>G_2</math>. Lines <math>AC</math> and <math>BD</math> meet at <math>E</math>; lines <math>AN</math> and <math>CD</math> meet at <math>P</math>; lines <math>BN</math> and <math>CD</math> meet at <math>Q</math>. Show that <math>EP=EQ</math>.
 
Two circles <math>G_1</math> and <math>G_2</math> intersect at two points <math>M</math> and <math>N</math>. Let <math>AB</math> be the line tangent to these circles at <math>A</math> and <math>B</math>, respectively, so that <math>M</math> lies closer to <math>AB</math> than <math>N</math>. Let <math>CD</math> be the line parallel to <math>AB</math> and passing through the point <math>M</math>, with <math>C</math> on <math>G_1</math> and <math>D</math> on <math>G_2</math>. Lines <math>AC</math> and <math>BD</math> meet at <math>E</math>; lines <math>AN</math> and <math>CD</math> meet at <math>P</math>; lines <math>BN</math> and <math>CD</math> meet at <math>Q</math>. Show that <math>EP=EQ</math>.
  
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[[2000 IMO Problems/Problem 1 | Solution]]
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[[2000 IMO Problems/Problem 1 | Solution]]
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== Problem 2 ==
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Let <math>a, b, c</math> be positive real numbers with <math>abc=1</math>. Show that
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<cmath>\left( a-1+\frac{1}{b} \right)\left( b-1+\frac{1}{c} \right)\left( c-1+\frac{1}{a} \right) \le 1</cmath>
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[[2000 IMO Problems/Problem 2 | Solution]]
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== Problem 3 ==
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Let <math>n \ge 2</math> be a positive integer and <math>\lambda</math> a positive real number. Initially there are <math>n</math> fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points <math>A</math> and <math>B</math> to the left of <math>B</math>, and letting the flea from <math>A</math> jump over the flea from <math>B</math> to the point <math>C</math> so that <math>\frac{BC}{AB}=\lambda</math>.
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Determine all values of <math>\lambda</math> such that, for any point <math>M</math> on the line and for any initial position of the <math>n</math> fleas, there exists a sequence of moves that will take them all to the position right of <math>M</math>.
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[[2000 IMO Problems/Problem 3 | Solution]]
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== Day 2 ==
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== Problem 4 ==
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A magician has one hundred cards numbered <math>1</math> to <math>100</math>. He puts them into three boxes,
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a red one, a white one and a blue one, so that each box contains at least one card.
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A member of the audience selects two of the three boxes, chooses one card from each
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and announces the sum of the numbers on the chosen cards. Given this sum, the magician
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identifies the box from which no card has been chosen.
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How many ways are there to put all the cards into the boxes so that this trick always
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works? (Two ways are considered different if at least one card is put into a different box.)
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[[2000 IMO Problems/Problem 4 | Solution]]
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== Problem 5 ==
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Does there exist a positive integer <math>n</math> such that <math>n</math> has exactly 2000 prime divisors and <math>n</math> divides <math>2^n+1</math>?
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[[2000 IMO Problems/Problem 5 | Solution]]
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== Problem 6 ==
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Let <math>\overline{AH_1}</math>, <math>\overline{BH_2}</math>, and <math>\overline{CH_3}</math> be the altitudes of an acute triangle <math>ABC</math>.  The incircle <math>\omega</math> of triangle <math>ABC</math> touches the sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>T_1</math>, <math>T_2</math>, and <math>T_3</math>, respectively.  Consider the reflections of the lines <math>H_1H_2</math>, <math>H_2H_3</math>, and <math>H_3H_1</math> with respect to the lines <math>T_1T_2</math>, <math>T_2T_3</math>, and <math>T_3T_1</math>.  Prove that these images form a triangle whose vertices line on <math>\omega</math>.
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[[2000 IMO Problems/Problem 6 | Solution]]

Latest revision as of 12:50, 19 April 2024

Day 1

Problem 1

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, so that $M$ lies closer to $AB$ than $N$. Let $CD$ be the line parallel to $AB$ and passing through the point $M$, with $C$ on $G_1$ and $D$ on $G_2$. Lines $AC$ and $BD$ meet at $E$; lines $AN$ and $CD$ meet at $P$; lines $BN$ and $CD$ meet at $Q$. Show that $EP=EQ$.

Solution


Problem 2

Let $a, b, c$ be positive real numbers with $abc=1$. Show that

\[\left( a-1+\frac{1}{b} \right)\left( b-1+\frac{1}{c} \right)\left( c-1+\frac{1}{a} \right) \le 1\]

Solution

Problem 3

Let $n \ge 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $A$ and $B$ to the left of $B$, and letting the flea from $A$ jump over the flea from $B$ to the point $C$ so that $\frac{BC}{AB}=\lambda$.

Determine all values of $\lambda$ such that, for any point $M$ on the line and for any initial position of the $n$ fleas, there exists a sequence of moves that will take them all to the position right of $M$.

Solution

Day 2

Problem 4

A magician has one hundred cards numbered $1$ to $100$. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.

A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.

How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)

Solution

Problem 5

Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $n$ divides $2^n+1$? Solution

Problem 6

Let $\overline{AH_1}$, $\overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute triangle $ABC$. The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ at $T_1$, $T_2$, and $T_3$, respectively. Consider the reflections of the lines $H_1H_2$, $H_2H_3$, and $H_3H_1$ with respect to the lines $T_1T_2$, $T_2T_3$, and $T_3T_1$. Prove that these images form a triangle whose vertices line on $\omega$.

Solution

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