Difference between revisions of "2015 UMO Problems"
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==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 | + | 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 |
− | + | <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 | + | 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 | + | |
− | 10 | + | <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 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 .
• The three greatest common divisors obtained in this way are all distinct.
What is the smallest possible value of ?
Problem 2
In and
, and the length of the altitude from
to
is
. If
is the midpoint of
, find all possible length(s) of
, and demonstrate that these
length(s) are achievable.
Problem 3
Find, with proof, all positive integers with
such that the greatest common divisor of the coefficients of
is equal to exactly
.
Problem 4
Anastasia and Balthazar need to go to the grocery store, which is km away. Anastasia
walks at
km/hr, and Balthazar walks at
km/hr. However, they also own a single bike, and
each of them bikes at
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?
Problem 5
A grid is filled with integers (positive or negative) such that the product of the integers
in any row or column is equal to
. For example, one possible grid is:
In how many ways can this be done?
Problem 6
A triangular pyramid with apex and base
has the property that the perimeter of
is
. Additionally, one can place a cylinder of radius
and height
completely inside the pyramid such that one of its bases is in the same plane as
.
What is the minimum possible height from
to apex
? Show that this height is achievable.