Difference between revisions of "1980 Canadian MO Problems"
Ellentilburg (talk | contribs) (→Problem 2) |
Ellentilburg (talk | contribs) (→Problem 4) |
||
| (4 intermediate revisions by the same user not shown) | |||
| Line 2: | Line 2: | ||
If <math>a679b</math> is the decimal expansion of a number in base <math>10</math>, such that it is divisible by <math>72</math>, determine <math> a,b</math>. | If <math>a679b</math> is the decimal expansion of a number in base <math>10</math>, such that it is divisible by <math>72</math>, determine <math> a,b</math>. | ||
| − | |||
[[1980 Canadian MO Problems/Problem 1 | Solution]] | [[1980 Canadian MO Problems/Problem 1 | Solution]] | ||
| Line 14: | Line 13: | ||
==Problem 3== | ==Problem 3== | ||
| − | + | A triangle is given with one constant angle and a constant inradius. Explicitly define all triangles for which the minimum perimeter is achieved. | |
[[1980 Canadian MO Problems/Problem 3 | Solution]] | [[1980 Canadian MO Problems/Problem 3 | Solution]] | ||
| Line 20: | Line 19: | ||
==Problem 4== | ==Problem 4== | ||
| + | A gambling student tosses a fair coin. She gains <math>1</math> point for each head that turns up, and gains <math>2</math> points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] <math>n</math> points is <math>\boxed{\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)}</math>. | ||
[[1980 Canadian MO Problems/Problem 4 | Solution]] | [[1980 Canadian MO Problems/Problem 4 | Solution]] | ||
| Line 25: | Line 25: | ||
==Problem 5== | ==Problem 5== | ||
| − | + | A parallelepiped has the property that all cross sections, which are parallel to any fixed face <math>F</math>, have the same perimeter as <math>F</math>. Determine whether or not any other poyhedron has this property. | |
[[1980 Canadian MO Problems/Problem 5 | Solution]] | [[1980 Canadian MO Problems/Problem 5 | Solution]] | ||
Latest revision as of 11:30, 2 March 2020
Problem 1
If 
is the decimal expansion of a number in base 
, such that it is divisible by 
, determine 
.
Problem 2
The numbers from 
to 
are printed on cards. The cards are shuffled and then laid out face up in 
rows of cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrrangement, do the cards in the rows still increase from left to right?
Problem 3
A triangle is given with one constant angle and a constant inradius. Explicitly define all triangles for which the minimum perimeter is achieved.
Problem 4
A gambling student tosses a fair coin. She gains point for each head that turns up, and gains 
points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] 
points is 
.
Problem 5
A parallelepiped has the property that all cross sections, which are parallel to any fixed face 
, have the same perimeter as . Determine whether or not any other poyhedron has this property.
