Difference between revisions of "2022 EGMO Problems"
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===Problem 6=== | ===Problem 6=== | ||
Let <math>ABCD</math> be a cyclic quadrilateral with circumcenter <math>O</math>. Let the internal angle bisectors at <math>A</math> and <math>B</math> meet at <math>X</math>, the internal angle bisectors at <math>B</math> and <math>C</math> meet at <math>Y</math>, the internal angle bisectors at <math>C</math> and <math>D</math> meet at <math>Z</math>, and the internal angle bisectors at <math>D</math> and <math>A</math> meet at <math>W</math>. Further, let <math>AC</math> and <math>BD</math> meet at <math>P</math>. Suppose that the points <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math>, <math>O</math>, and <math>P</math> are distinct. | Let <math>ABCD</math> be a cyclic quadrilateral with circumcenter <math>O</math>. Let the internal angle bisectors at <math>A</math> and <math>B</math> meet at <math>X</math>, the internal angle bisectors at <math>B</math> and <math>C</math> meet at <math>Y</math>, the internal angle bisectors at <math>C</math> and <math>D</math> meet at <math>Z</math>, and the internal angle bisectors at <math>D</math> and <math>A</math> meet at <math>W</math>. Further, let <math>AC</math> and <math>BD</math> meet at <math>P</math>. Suppose that the points <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math>, <math>O</math>, and <math>P</math> are distinct. | ||
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Prove that <math>O</math>, <math>X</math>, <math>Y</math> <math>Z</math>, <math>W</math> lie on the same circle if and only if <math>P</math>, <math>X</math>, <math>Y</math>, <math>Z</math>, and <math>W</math> lie on the same circle. | Prove that <math>O</math>, <math>X</math>, <math>Y</math> <math>Z</math>, <math>W</math> lie on the same circle if and only if <math>P</math>, <math>X</math>, <math>Y</math>, <math>Z</math>, and <math>W</math> lie on the same circle. | ||
[[2022 EGMO Problems/Problem 6|Solution]] | [[2022 EGMO Problems/Problem 6|Solution]] |
Latest revision as of 14:13, 24 December 2022
Contents
Day 1
Problem 1
Let be an acute-angled triangle in which
and
. Let point
lie on segment
and point
lie on segment
such that
,
and
. Let
be the circumcenter of triangle
,
the orthocenter of triangle
, and
the point of intersection of the lines
and
. Prove that
,
, and
are collinear.
Problem 2
Let be the set of all positive integers. Find all functions
such that for any positive integers
and
, the following two conditions hold:
(i)
, and
(ii) at least two of the numbers ,
, and
are equal.
Problem 3
An infinite sequence of positive integers is called
if
(i) is a perfect square, and
(ii) for any integer ,
is the smallest positive integer such that
is a perfect square.
Prove that for any good sequence , there exists a positive integer
such that
for all integers
.
Day 2
Problem 4
Given a positive integer , determine the largest positive integer
for which there exist
real numbers
such that
and
for
.
Problem 5
For all positive integers ,
, let
be the number of ways an
board can be fully covered by
dominoes of size
. (For example,
and
.) Find all positive integers
such that for every positive integer
, the number
is odd.
Problem 6
Let be a cyclic quadrilateral with circumcenter
. Let the internal angle bisectors at
and
meet at
, the internal angle bisectors at
and
meet at
, the internal angle bisectors at
and
meet at
, and the internal angle bisectors at
and
meet at
. Further, let
and
meet at
. Suppose that the points
,
,
,
,
, and
are distinct.
Prove that ,
,
,
lie on the same circle if and only if
,
,
,
, and
lie on the same circle.