MathDB

3

Part of 2015 Caucasus Mathematical Olympiad

Problems(5)

cupcakes and bagels for Petya, Apya and Kolya

Source: Caucasus 7.3

4/26/2019
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
system of equationsalgebra
same no of tiles of 2x2 and 3x1 tile completely a floor nxn

Source: Caucasus 2015 8.3

4/26/2019
The workers laid a floor of size n×nn \times n with tiles of two types: 2×22 \times 2 and 3×13 \times 1. It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen? (You can’t cut tiles and also put them on top of each other.)
Tilingtilescombinatoricscombinatorial geometry
if <BLA= <BAC$, then BP = CP (Caucasus 2015 geometry for 9th grade)

Source: I Caucasus 2015 9.3

9/6/2018
Let ALAL be the angle bisector of the acute-angled triangle ABCABC. and ω\omega be the circle circumscribed about it. Denote by PP the intersection point of the extension of the altitude BHBH of the triangle ABCABC with the circle ω\omega . Prove that if ∠BLA=∠BAC\angle BLA= \angle BAC, then BP=CPBP = CP.
geometryequal anglesequal segmentscircumcircleangle bisector
smallest no of 3-cell to point a 5x5 square, painting at most 1 corner

Source: Caucasus 2015 10.3

4/26/2019
What is the smallest number of 33-cell corners that you need to paint in a 5×55 \times5 square so that you cannot paint more than one corner of one it? (Shaded corners should not overlap.)
combinatoricscombinatorial geometryTilingtiles
same no of tiles of 2x2 and 5x1 tile completely a floor nxn , 10<n<20

Source: Caucasus 2015 11.3

4/26/2019
The workers laid a floor of size n×nn\times n (10<n<2010 <n <20) with two types of tiles: 2×22 \times 2 and 5×15\times 1. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which nn could this happen? (You can’t cut tiles and also put them on top of each other.)
combinatoricscombinatorial geometrytilesTiling