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Difference between revisions of "2016 IMO Problems"

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==Problem 3==
 
==Problem 3==
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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 integral 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>.
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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 08:48, 20 December 2023

Problem 1

Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.

Solution

Problem 2

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:

  • in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and
  • in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.

Note. The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

Solution

Problem 3

Let $P = A_1A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1,A_2,\dots, A_k$ have integer coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

Solution

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 $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $\{P(a+1),P(a+2),\ldots,P(a+b)\}$ is fragrant?

Solution

Problem 5

The equation

$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$

is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Solution

Problem 6

There are $n\ge 2$ 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 $n-1$ 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 $n$ is odd.

(b) Prove that Geoff can never fulfill his wish if $n$ is even.

Solution

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
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