Difference between revisions of "2016 IMO Problems"
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==Problem 3== | ==Problem 3== | ||
− | Let <math>P = A_1A_2 \cdots A_k</math> be a convex polygon in the plane. The vertices <math>A_1,A_2,\dots, A_k</math> have | + | Let <math>P = A_1A_2 \cdots A_k</math> be a convex polygon in the plane. The vertices <math>A_1,A_2,\dots, A_k</math> have integer coordinates and lie on a circle. Let <math>S</math> be the area of <math>P</math>. An odd positive integer <math>n</math> is given such that the squares of the side lengths of <math>P</math> are integers divisible by <math>n</math>. Prove that <math>2S</math> is an integer divisible by <math>n</math>. |
[[2016 IMO Problems/Problem 3|Solution]] | [[2016 IMO Problems/Problem 3|Solution]] | ||
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[[2016 IMO Problems/Problem 6|Solution]] | [[2016 IMO Problems/Problem 6|Solution]] | ||
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+ | {{IMO box|year=2016|before=[[2015 IMO Problems]]|after=[[2017 IMO Problems]]}} |
Latest revision as of 09:48, 20 December 2023
Problem 1
Triangle has a right angle at
. Let
be the point on line
such that
and
lies between
and
. Point
is chosen so that
and
is the bisector of
. Point
is chosen so that
and
is the bisector of
. Let
be the midpoint of
. Let
be the point such that
is a parallelogram. Prove that
and
are concurrent.
Problem 2
Find all integers for which each cell of
table can be filled with one of the letters
and
in such a way that:
|
Note. The rows and columns of an table are each labelled
to
in a natural order. Thus each cell corresponds to a pair of positive integer
with
. For
, the table has
diagonals of two types. A diagonal of first type consists all cells
for which
is a constant, and the diagonal of this second type consists all cells
for which
is constant.
Problem 3
Let be a convex polygon in the plane. The vertices
have integer coordinates and lie on a circle. Let
be the area of
. An odd positive integer
is given such that the squares of the side lengths of
are integers divisible by
. Prove that
is an integer divisible by
.
Problem 4
A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let . What is the least possible positive integer value of
such that there exists a non-negative integer
for which the set
is fragrant?
Problem 5
The equation
![$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$](http://latex.artofproblemsolving.com/7/4/0/740a0adcfbdff44b7db27d0b8974a6369d9948d2.png)
is written on the board, with linear factors on each side. What is the least possible value of
for which it is possible to erase exactly
of these
linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Problem 6
There are line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands
times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if is odd.
(b) Prove that Geoff can never fulfill his wish if is even.
2016 IMO (Problems) • Resources | ||
Preceded by 2015 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2017 IMO Problems |
All IMO Problems and Solutions |