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Difference between revisions of "Olimpiada de Mayo"
(→‎Past Tests/Results)
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* Venezuela
 
* Venezuela
 
==Past Tests/Results==
 
==Past Tests/Results==
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* The [http://www.oma.org.ar/enunciados/may3.htm 1997 test], [http://www.oma.org.ar/internacional/resultados-may3.htm 1997 test results]
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* The [http://www.oma.org.ar/enunciados/may3.htm 1997 test]
 
* The [http://www.oma.org.ar/enunciados/may4.htm 1998 test], [http://www.oma.org.ar/internacional/resultados-may4.htm 1998 test results]
 
* The [http://www.oma.org.ar/enunciados/may4.htm 1998 test], [http://www.oma.org.ar/internacional/resultados-may4.htm 1998 test results]
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* The [http://www.oma.org.ar/enunciados/may5.htm 1999 test], [http://www.oma.org.ar/internacional/resultados-may5.htm 1999 test results]
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* The [http://www.oma.org.ar/enunciados/may5.htm 1999 test]
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* The [http://www.oma.org.ar/enunciados/may6.htm 2000 test], [http://www.oma.org.ar/internacional/resultados-may6.htm 2000 test results]
+
* The [http://www.oma.org.ar/enunciados/may6.htm 2000 test]
−
* The [http://www.oma.org.ar/enunciados/may7.htm 2001 test], [http://www.oma.org.ar/internacional/resultados-may7.htm 2001 test results]
+
* The [http://www.oma.org.ar/enunciados/may7.htm 2001 test]
−
* The [http://www.oma.org.ar/enunciados/may8.htm 2002 test], [http://www.oma.org.ar/internacional/resultados-may8.htm 2002 test results]
+
* The [http://www.oma.org.ar/enunciados/may8.htm 2002 test]
−
* The [http://www.oma.org.ar/enunciados/may9.htm 2003 test], [http://www.oma.org.ar/internacional/resultados-may9.htm 2003 test results]
+
* The [http://www.oma.org.ar/enunciados/may9.htm 2003 test]
−
* The [http://www.oma.org.ar/enunciados/may10.htm 2004 test], [http://www.oma.org.ar/internacional/resultados-may10.htm 2004 test results]
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* The [http://www.oma.org.ar/enunciados/may10.htm 2004 test]
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* The [http://www.oma.org.ar/enunciados/may11.htm 2005 test], [http://www.oma.org.ar/internacional/resultados-may11.htm 2005 test results]
+
* The [http://www.oma.org.ar/enunciados/may11.htm 2005 test]
 
* The [http://www.oma.org.ar/enunciados/may12.htm 2006 test], [http://www.oma.org.ar/internacional/resultados-may12.htm 2006 test results]
 
* The [http://www.oma.org.ar/enunciados/may12.htm 2006 test], [http://www.oma.org.ar/internacional/resultados-may12.htm 2006 test results]
 
* The [http://www.oma.org.ar/enunciados/may13.htm 2007 test], [http://www.oma.org.ar/internacional/resultados-may13.htm 2007 test results]
 
* The [http://www.oma.org.ar/enunciados/may13.htm 2007 test], [http://www.oma.org.ar/internacional/resultados-may13.htm 2007 test results]
 
* The [http://www.oma.org.ar/enunciados/may14.htm 2008 test], [http://www.oma.org.ar/internacional/resultados-may14.htm 2008 test results]
 
* The [http://www.oma.org.ar/enunciados/may14.htm 2008 test], [http://www.oma.org.ar/internacional/resultados-may14.htm 2008 test results]
 +
 
==Problems==
 
==Problems==
 
Some problems from previous Olimpiada de Mayo tests.
 
Some problems from previous Olimpiada de Mayo tests.

Revision as of 18:00, 20 May 2009

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 $13$ years the year before, and level two is for students that have not reached the age of $15$ years the year before. The test is taken by $12$ latinamerican countries.

Member Countries

  • Argentina
  • Brasil
  • Bolivia
  • Colombia
  • Ecuador
  • El Salvador
  • México
  • Panamá
  • Paraguay
  • Perú
  • Puerto Rico
  • Venezuela

Past Tests/Results

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 $388$ and end in $388$? 2nd level, 3rd olympiad test

Solutions

Solution to Problem #1

Let the two numbers be $a$ and $b$ such that $a > b$.

We know that $a - b = 7$ and $ab\equiv 0\pmod{5}$, since only one of the two numbers can be a multiple of $5$ we can start with cases, one in which $a$ is a multiple of $5$ and the other where $b$ is a multiple of $5$.

Case 1: ($a$ is a multiple of $5$) 

The lowest value for $a$ that satisfies the conditions is $10 = 5(2)$.

The highest value for $a$ that satisfies the conditions is $100 = 5(20)$.

So there are a total of $20 - 2 + 1 = 19$ cases that work.

Case 2: ($b$ is a multiple of $5$) 

The lowest value for $b$ that satisfies the conditions is $5=5(1)$.

The highest value for $b$ that satisfies the conditions is $90 = 5(18)$.

So there are a total of $18$ cases that work.


In total there are $19 + 18 = \boxed{37}$ cases that work.

Solution to Problem #2

First we look for the Least common multiple of $388$ that ends in $3$ zeros so that when added to a number that ends in $388$ it will still be a multiple of $388$ and end in $388$, and this number is $97000=97\cdot 1000=388\cdot 25$.

So any number in the form of $3880388 + 97000k$ ends in $388$ and is a multiple of $388$.

We want to find $1000000 < 3880388 + 97000k < 10000000$

Since we are only dealing with the thousands we don't need to worry about the $388$:

$\begin{eqnarray*} 1000 & < & 3880 + 97k < 100000 \\ 1000 & < & 97(40 + k) < 10000 \\ 10 & < & 40 + k < 103 \\ - 30 & < & k < 63 \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

From this we see that there are $\boxed{92}$ solutions.

Reference

Olimpiada de Mayo home

Practice tests

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