SA Mathematics Olympiad (2015) - 3 Tough Questions

The first round of the South African Mathematics Olympiad was held today at various high schools throughout the country. Can you solve the two hardest questions in the paper?

Note by Mark Mottian
4 years, 6 months ago

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Solution to QUESTION 19 : ( Its not very complicated)

Let FECG be a kite , then

FE=FGandEC=GCFE=FG and EC=GC

GC=GAGC=GA (Since diagonals of a rhombus bisect each other)

EC=BEEC=BE (Since EE is mid-point of BSBS

Therefore, BC=ACBC=AC

But, AB=BCAB=BC (Sides of a rhombus)

So, AB=BC=ACAB=BC=AC

ABC\bigtriangleup ABC is an equilateral triangle

Since medians of an equilateral triangle also bisect the angle from which they are drawn,

2x=602x = 60

x=30x=\boxed{30}

Rishabh Tripathi - 4 years, 6 months ago

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A really good solution! Good job!

Mark Mottian - 4 years, 6 months ago

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This challenging question also featured in the paper:

Triangle ABC has a point D on line AB such that D is the midpoint of AB. E is the midpoint of CD. F is the midpoint of AE. If the area of triangle ABC is 24, find the area of triangle DEF.

(A)   3
(B)   4
(C)   6
(D)   8
(E)   9

Mark Mottian - 4 years, 6 months ago

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Repeatedly using the fact that the midpoint of a triangle divides the triangle into 2 2 triangles of equal area should get you the answer.

Hijacking @@Krishna Sharma's diagram, we have [ADC]=12[ABC] [ADC] = \frac{1}{2}[ABC] , [ADE]=12[ADC][ADE] = \frac{1}{2}[ADC] and [DEF]=12[ADE] [DEF] = \frac{1}{2}[ADE] . Combining all the equations, we have [DEF]=18[ABC]=3 [DEF] = \frac{1}{8}[ABC] = 3

Siddhartha Srivastava - 4 years, 6 months ago

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Hi Siddhartha! This is a very insightful solution. If you knew that the "median of the triangle divides the triangle into two triangles of equal area", it makes the problem very easy. Your solution is definitely elegant. Thank you so much for sharing!

Mark Mottian - 4 years, 6 months ago

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yupp.. nice and easy.. without complications

Rishabh Tripathi - 4 years, 6 months ago

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The answer is 3. Since nothing is given for simplicity consider the triangle is equilateral,find out the the square of side(yes side length not required), Now agian for simplicity consider the AB as base of triangle with vertex A at origin and we are done just find out coordinates of D,E,F in terms of 'a' and you are done!

image image

Sorry for my poor handwriting :p

Krishna Sharma - 4 years, 6 months ago

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For 20, I don't see any easier solution other than seeing that a rectangular prism with side lengths 6,5,4 6,5,4 works and thus the surface area is 148 148 .

Siddhartha Srivastava - 4 years, 6 months ago

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Siddhartha, how did you figure out that the side lengths were 4,5 and 6?

Mark Mottian - 4 years, 6 months ago

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Pretty much a lucky guess. We see that its the only solution to xyz=120 xyz = 120 and (x2)(y2)(z2)=24 (x-2)(y-2)(z-2) = 24 .

The first equation is obvious. The second one comes is from the fact that the total volume of the cuboid not colored is (x2)(y2)(z2) (x-2)(y-2)(z-2) , since we have to discount the cubes which appear on either ends.

Siddhartha Srivastava - 4 years, 6 months ago

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I think the answer of Q.19 is c) 30°.

Dipesh Shivrame - 4 years, 6 months ago

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