Difference between revisions of "1997 JBMO Problems"
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==Problem 1== | ==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 <math>\frac{1}{8}</math> | + | 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]] | |
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| + | |||
| + | Hyy I am anshu Kumar anahu from https://t.me/AOPS01 | ||
==Problem 2== | ==Problem 2== | ||
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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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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== | ||
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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]] | ||
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| + | ==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 
.
Hyy I am anshu Kumar anahu from https://t.me/AOPS01
Problem 2
Let 
. Compute the following expression in terms of 
:
![\[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\]](http://latex.artofproblemsolving.com/c/9/9/c99e935bc847cdcaee76b4a8af84db262067eca7.png)
Problem 3
Let 
be a triangle and let 
be the incenter. Let 
, 
be the midpoints of the sides 
and 
respectively. The lines 
and 
meet 
at 
and 
respectively. Prove that 
.
Problem 4
Determine the triangle with sides 
and circumradius 
for which 
.
Problem 5
Let 
, 
, 
, 
be positive integers such that ![\[n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2.\]](http://latex.artofproblemsolving.com/2/7/9/27994bc5861679d68c68e81da1b333f9770e3752.png)
Show that at least two of the numbers are even.
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 | ||
