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Difference between revisions of "2015 UMO Problems"

(Created page with "==Problem 1== Three trolls have divided <math>n</math> pancakes among themselves such that: • Each troll has a positive integer number of pancakes. • The greatest common d...")
 
 
Line 1: Line 1:
 
==Problem 1==
 
==Problem 1==
 
Three trolls have divided <math>n</math> pancakes among themselves such that:
 
Three trolls have divided <math>n</math> pancakes among themselves such that:
 +
 
• Each troll has a positive integer number of pancakes.
 
• Each troll has a positive integer number of pancakes.
 +
 
• The greatest common divisor of the number of pancakes held by any two trolls is bigger than <math>1</math>.
 
• The greatest common divisor of the number of pancakes held by any two trolls is bigger than <math>1</math>.
 +
 
• The three greatest common divisors obtained in this way are all distinct.
 
• The three greatest common divisors obtained in this way are all distinct.
 +
 
What is the smallest possible value of <math>n</math>?
 
What is the smallest possible value of <math>n</math>?
  
Line 9: Line 13:
  
 
==Problem 2==
 
==Problem 2==
−
In <math>\triangle{ABC}, AC = 13 & AB = 20</math>, and the length of the altitude from A to
+
In <math>\triangle{ABC}, \overline{AC} = 13</math> and <math>\overline{AB} = 20</math>, and the length of the altitude from <math>A</math> to
−
←→BC is <math>12</math>. If <math>M</math> is the midpoint of <math>BC</math>, find all possible length(s) of <math>AM</math>, and demonstrate that these
+
<math>\overline{BC}</math> is <math>12</math>. If <math>M</math> is the midpoint of <math>\overline{BC}</math>, find all possible length(s) of <math>\overline{AM}</math>, and demonstrate that these
 
length(s) are achievable.
 
length(s) are achievable.
  
Line 17: Line 21:
 
==Problem 3==
 
==Problem 3==
  
−
Find, with proof, all positive integers <math>n</math> with <math>2 ≤ n ≤ 20</math> such that the greatest common divisor of the coefficients of <math>(x+y)^n-x^n-y^n</math>
+
Find, with proof, all positive integers <math>n</math> with <math>2\le n\le 20</math> such that the greatest common divisor of the coefficients of <math>(x+y)^n-x^n-y^n</math>
 
is equal to exactly <math>3</math>.
 
is equal to exactly <math>3</math>.
  
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A <math>3 \times 3</math> grid is filled with integers (positive or negative) such that the product of the integers
 
A <math>3 \times 3</math> grid is filled with integers (positive or negative) such that the product of the integers
 
in any row or column is equal to <math>20</math>. For example, one possible grid is:
 
in any row or column is equal to <math>20</math>. For example, one possible grid is:
−
1 −5 −4
+
 
−
10 −2 −1
+
<math>\begin{bmatrix}
−
2 2 5
+
  1 & -5& -4 \\
 +
    10 & -2 &-1 \\
 +
  2& 2& 5
 +
\end{bmatrix}</math>
  
 
In how many ways can this be done?
 
In how many ways can this be done?
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==Problem 6==
 
==Problem 6==
  
−
A triangular pyramid with apex <math>O</math> and base <math>ABC</math> has the property that the perimeter of <math>\triangle {ABC}</math> is <math>84</math>. Additionally, one can place a cylinder of radius <math>4</math> and height <math>10</math> completely inside the pyramid such that one of its bases is in the same plane as <math>\triangle{ABC}</math>.
+
A triangular pyramid with apex <math>O</math> and base <math>\triangle{ABC}</math> has the property that the perimeter of <math>\triangle {ABC}</math> is <math>84</math>. Additionally, one can place a cylinder of radius <math>4</math> and height <math>10</math> completely inside the pyramid such that one of its bases is in the same plane as <math>\triangle{ABC}</math>.
 
What is the minimum possible height from <math>\triangle{ABC}</math> to apex <math>O</math>? Show that this height is achievable.
 
What is the minimum possible height from <math>\triangle{ABC}</math> to apex <math>O</math>? Show that this height is achievable.
  
 
[[2015 UMO Problems/Problem 6|Solution]]
 
[[2015 UMO Problems/Problem 6|Solution]]

Latest revision as of 01:51, 6 November 2015

Problem 1

Three trolls have divided $n$ pancakes among themselves such that:

• Each troll has a positive integer number of pancakes.

• The greatest common divisor of the number of pancakes held by any two trolls is bigger than $1$.

• The three greatest common divisors obtained in this way are all distinct.

What is the smallest possible value of $n$?

Solution

Problem 2

In $\triangle{ABC}, \overline{AC} = 13$ and $\overline{AB} = 20$, and the length of the altitude from $A$ to $\overline{BC}$ is $12$. If $M$ is the midpoint of $\overline{BC}$, find all possible length(s) of $\overline{AM}$, and demonstrate that these length(s) are achievable.

Solution

Problem 3

Find, with proof, all positive integers $n$ with $2\le n\le 20$ such that the greatest common divisor of the coefficients of $(x+y)^n-x^n-y^n$ is equal to exactly $3$.

Solution

Problem 4

Anastasia and Balthazar need to go to the grocery store, which is $100$ km away. Anastasia walks at $5$ km/hr, and Balthazar walks at $4$ km/hr. However, they also own a single bike, and each of them bikes at $10$ km/hr. They are allowed to go forwards or backwards, and the bike will not get stolen if they drop it off along the way for the other person to pick up. What is the shortest amount of time necessary for both of them to get to the grocery store?

Solution

Problem 5

A $3 \times 3$ grid is filled with integers (positive or negative) such that the product of the integers in any row or column is equal to $20$. For example, one possible grid is:

$\begin{bmatrix} 1 & -5& -4 \\ 10 & -2 &-1 \\ 2& 2& 5 \end{bmatrix}$

In how many ways can this be done?

Solution

Problem 6

A triangular pyramid with apex $O$ and base $\triangle{ABC}$ has the property that the perimeter of $\triangle {ABC}$ is $84$. Additionally, one can place a cylinder of radius $4$ and height $10$ completely inside the pyramid such that one of its bases is in the same plane as $\triangle{ABC}$. What is the minimum possible height from $\triangle{ABC}$ to apex $O$? Show that this height is achievable.

Solution

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