Latvian Math Olympiad (2011), Problems 5.1-5.5

Problems and solutions of Latvian Open Math Olympiad, 2011. All 5 problems are for grade 5 students (12 year olds). The problemset is for 5 hours. Any child in Latvia can participate - hence it is "open". More info - http://nms.lu.lv/ .
Problemset in Latvian - http://nms.lu.lv/olimpiades/aol/10_11/ao38_uzd.pdf ; solutions in Latvian - http://nms.lu.lv/olimpiades/aol/10_11/ao38_atr.pdf .


== Problem 5.1 ==
In the multiplication example AB * CD = EEE digits
were replaced with letters (equal digits are represented
by equal letters; different digits -- by different letters),
and neither A nor C is 0.
Restore the original multiplication example. Find all
possible solutions!


== Problem 5.2 ==
In the 3x3 table each cell should contain a positive integer
so that every row, every column and both diagonals add
up to the same result. 3 of the positive integers are known
(see figure). Fill in the others!
(Figure)
(?|15|?)
(?|11|?)
(?|?|18)


== Problem 5.3 ==
Show how a square can be cut into obtuse triangles!


== Problem 5.4 ==
Is it possible to write integer numbers from 1 to 12
in some order in a circle (using each number exactly once)
so that the difference between any two neighboring
numbers on the circle is
(a) either 2 or 3
(b) either 3 or 4


== Problem 5.5 ==
On a square grid we draw a square 7x7
(i.e. a shape enclosing 7 by 7 small squares from the grid).
On this square we should place n "corners"
(shapes having 3 squares each - see the figure) so that
it would not be possible to place any other such "corner".
"Corners" cannot overlap, their borders should coincide
with the grid lines, "corners" can also be rotated.
Show how it can be done if
(a) n = 9;
(b) n = 8.
(Figure)
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Category:

Education

Tags:

• math
• olympiad
• training

[blockman11]

This video was great, it really helped reveal a couple new tricks which will be useful for future math competitions. Thanks!!!