Difference between revisions of "2016 JBMO Problems"
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==Problem 1== | ==Problem 1== | ||
| − | A trapezoid <math>ABCD</math> (<math>AB || | + | A trapezoid <math>ABCD</math> (<math>AB || CD</math>,<math>AB > CD</math>) is circumscribed.The incircle of the triangle <math>ABC</math> touches the lines <math>AB</math> and <math>AC</math> at the points <math>M</math> and <math>N</math>,respectively.Prove that the incenter of the trapezoid <math>ABCD</math> lies on the line <math>MN</math>. |
[[2016 JBMO Problems/Problem 1#Solution|Solution]] | [[2016 JBMO Problems/Problem 1#Solution|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
| + | |||
| + | Let <math>a,b,c</math> be positive real numbers.Prove that | ||
| + | |||
| + | <math>\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}</math>. | ||
[[2016 JBMO Problems/Problem 2#Solution|Solution]] | [[2016 JBMO Problems/Problem 2#Solution|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
| + | |||
| + | Find all triplets of integers <math>(a,b,c)</math> such that the number | ||
| + | <cmath>N = \frac{(a-b)(b-c)(c-a)}{2} + 2</cmath> | ||
| + | is a power of <math>2016</math>. | ||
| + | |||
| + | (A power of <math>2016</math> is an integer of form <math>2016^n</math>,where <math>n</math> is a non-negative integer.) | ||
[[2016 JBMO Problems/Problem 3#Solution|Solution]] | [[2016 JBMO Problems/Problem 3#Solution|Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
| + | A <math>5 \times 5</math> table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every <math>2 \times 2</math> subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count. | ||
[[2016 JBMO Problems/Problem 4#Solution|Solution]] | [[2016 JBMO Problems/Problem 4#Solution|Solution]] | ||
Latest revision as of 07:53, 22 June 2024
Problem 1
A trapezoid 
(
,
) is circumscribed.The incircle of the triangle 
touches the lines 
and 
at the points 
and 
,respectively.Prove that the incenter of the trapezoid lies on the line 
.
Problem 2
Let 
be positive real numbers.Prove that
.
Problem 3
Find all triplets of integers 
such that the number
![\[N = \frac{(a-b)(b-c)(c-a)}{2} + 2\]](http://latex.artofproblemsolving.com/4/9/6/496f7460105427895e3c88cb3d99ffadfb21bc41.png)
is a power of 
.
(A power of is an integer of form 
,where 
is a non-negative integer.)
Problem 4
A 
table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every 
subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
See also
| 2016 JBMO (Problems • Resources) | ||
| Preceded by 2015 JBMO Problems |
Followed by 2017 JBMO Problems | |
| 1 • 2 • 3 • 4 | ||
| All JBMO Problems and Solutions | ||
