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Difference between revisions of "1997 JBMO Problems"

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==Problem 1==
 
==Problem 1==
  
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''(Bulgaria)'' Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than <math>\frac{1}{8}</math>
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Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than <math>\frac{1}{8}</math>.
 +
 
 +
[[1997 JBMO Problems/Problem 1#Solution|Solution]]
 +
 
 +
 
 +
Hyy I am anshu Kumar anahu from https://t.me/AOPS01
  
 
==Problem 2==
 
==Problem 2==
  
−
''(Cyprus)'' Let <math>\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k</math>. Compute the following expression in terms of <math>k</math>:  
+
Let <math>\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k</math>. Compute the following expression in terms of <math>k</math>:  
 
<cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.  </cmath>
 
<cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.  </cmath>
 +
 +
[[1997 JBMO Problems/Problem 2#Solution|Solution]]
  
 
==Problem 3==
 
==Problem 3==
  
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''(Greece)'' Let <math>ABC</math> be a triangle and let <math>I</math> be the incenter. Let <math>N</math>, <math>M</math> be the midpoints of the sides <math>AB</math> and <math>CA</math> respectively. The lines <math>BI</math> and <math>CI</math> meet <math>MN</math> at <math>K</math> and <math>L</math> respectively. Prove that <math>AI+BI+CI>BC+KL</math>.
+
Let <math>ABC</math> be a triangle and let <math>I</math> be the incenter. Let <math>N</math>, <math>M</math> be the midpoints of the sides <math>AB</math> and <math>CA</math> respectively. The lines <math>BI</math> and <math>CI</math> meet <math>MN</math> at <math>K</math> and <math>L</math> respectively. Prove that <math>AI+BI+CI>BC+KL</math>.
 +
 
 +
[[1997 JBMO Problems/Problem 3#Solution|Solution]]
  
 
==Problem 4==
 
==Problem 4==
  
−
''(Romania)'' Determine the triangle with sides <math>a,b,c</math> and circumradius <math>R</math> for which <math>R(b+c) = a\sqrt{bc}</math>.  
+
Determine the triangle with sides <math>a,b,c</math> and circumradius <math>R</math> for which <math>R(b+c) = a\sqrt{bc}</math>.  
 +
 
 +
[[1997 JBMO Problems/Problem 4#Solution|Solution]]
  
 
==Problem 5==
 
==Problem 5==
  
 
Let <math>n_1</math>, <math>n_2</math>, <math>\ldots</math>, <math>n_{1998}</math> be positive integers such that <cmath> n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2. </cmath> Show that at least two of the numbers are even.
 
Let <math>n_1</math>, <math>n_2</math>, <math>\ldots</math>, <math>n_{1998}</math> be positive integers such that <cmath> n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2. </cmath> Show that at least two of the numbers are even.
 +
 +
[[1997 JBMO Problems/Problem 5#Solution|Solution]]
 +
 +
==See also==
 +
{{JBMO box|year=1997|before=First Olympiad|after=[[1998 JBMO Problems]]}}

Latest revision as of 21:44, 14 September 2023

Problem 1

Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than $\frac{1}{8}$.

Solution


Hyy I am anshu Kumar anahu from https://t.me/AOPS01

Problem 2

Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$: \[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\]

Solution

Problem 3

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$.

Solution

Problem 4

Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$.

Solution

Problem 5

Let $n_1$, $n_2$, $\ldots$, $n_{1998}$ be positive integers such that \[n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2.\] Show that at least two of the numbers are even.

Solution

See also

1997 JBMO (Problems • Resources)
Preceded by
First Olympiad 		Followed by
1998 JBMO Problems
1 • 2 • 3 • 4 • 5
All JBMO Problems and Solutions


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