Difference between revisions of "Olimpiada de Mayo"
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* Bolivia | * Bolivia | ||
* Colombia | * Colombia | ||
| + | * Costa Rica | ||
* Ecuador | * Ecuador | ||
* El Salvador | * El Salvador | ||
| + | * España | ||
* México | * México | ||
* Panamá | * Panamá | ||
* Paraguay | * Paraguay | ||
* Perú | * Perú | ||
| + | * Portugal | ||
* Puerto Rico | * Puerto Rico | ||
* Venezuela | * Venezuela | ||
| + | |||
==Past Tests/Results== | ==Past Tests/Results== | ||
* The [http://www.oma.org.ar/enunciados/may3.htm 1997 test] | * The [http://www.oma.org.ar/enunciados/may3.htm 1997 test] | ||
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Since we are only dealing with the thousands we don't need to worry about the <math>388</math>: | Since we are only dealing with the thousands we don't need to worry about the <math>388</math>: | ||
| − | < | + | <cmath>\begin{eqnarray*} 1000 & < & 3880 + 97k < 100000 \\ |
1000 & < & 97(40 + k) < 10000 \\ | 1000 & < & 97(40 + k) < 10000 \\ | ||
10 & < & 40 + k < 103 \\ | 10 & < & 40 + k < 103 \\ | ||
| − | - 30 & < & k < 63 \end{eqnarray*}</ | + | - 30 & < & k < 63 \end{eqnarray*}</cmath> |
From this we see that there are <math>\boxed{92}</math> solutions. | From this we see that there are <math>\boxed{92}</math> solutions. | ||
Latest revision as of 13:52, 17 August 2020
Olimpiada de Mayo(May Olympiad) is an annual test that has two levels, level one is for students that have not reached the age of 
years the year before, and level two is for students that have not reached the age of 
years the year before. The test is taken by 
latinamerican countries.
Contents
Member Countries
- Argentina
- Brasil
- Bolivia
- Colombia
- Costa Rica
- Ecuador
- El Salvador
- España
- México
- Panamá
- Paraguay
- Perú
- Portugal
- Puerto Rico
- Venezuela
Past Tests/Results
- The 1997 test
- The 1998 test, 1998 test results
- The 1999 test
- The 2000 test
- The 2001 test
- The 2002 test
- The 2003 test
- The 2004 test
- The 2005 test
- The 2006 test, 2006 test results
- The 2007 test, 2007 test results
- The 2008 test, 2008 test results
- The 2009 test, 2009 test results
Sample Problems
Some problems from previous Olimpiada de Mayo tests.
Problem 1
We choose 2 numbers in between 1 and 100, inclusive, such that their difference is 7 and their product is a multiple of 5. How many ways can we do this? 1st level, 5th olympiad test
Problem 2
How many 7-digit numbers are multiples of 
and end in ? 2nd level, 3rd olympiad test
Solutions
Solution to Problem #1
Let the two numbers be 
and 
such that 
.
We know that 
and 
, since only one of the two numbers can be a multiple of 
we can start with cases, one in which is a multiple of and the other where is a multiple of .
Case 1: ( is a multiple of )
The lowest value forthat satisfies the conditions is
.
.
The highest value for.
.
So there are a total of 
cases that work.
Case 2: ( is a multiple of )
The lowest value forthat satisfies the conditions is
.
.
The highest value for.
.
So there are a total of 
cases that work.
In total there are 
cases that work.
Solution to Problem #2
First we look for the Least common multiple of that ends in 
zeros so that when added to a number that ends in it will still be a multiple of and end in , and this number is 
.
So any number in the form of 
ends in and is a multiple of .
We want to find 
Since we are only dealing with the thousands we don't need to worry about the :
From this we see that there are 
solutions.
