Tegi 2016

AVTORJA: Gregor Kikelj, T2C, Domen Težak, T2C
PRILOGA
NASLOV
Mathematical Problems
Test your skill
1. Consider a circle with a point on its circumference. It is split in one section only. Draw another
point on the circumference and connect it with the original point. Now the circle is split into 2
parts. Add another point and connect it with the previous.
The picture demonstrates that
with 4 points the answer is 8
since there are 8 sections.
The circle is now split into 4 parts. Repeat the process,
until you get 10 points all connected to each other.
Also no 3 connections may intersect in a single point.
How many parts is the circle split on now after adding
the 10th point?
Solution 1: If we add the points where the lines intersect
we get a structure called graph. For every graph the
equation V-E+F=2 holds where V is the number of verticals
or points, E is the number of edges or lines and F is
the number of areas or faces. The equation clearly implies
F=2+E-V. Let's count the points now(we defined intersections
as points). There are n points on the circumference
and some inside. Every inside point is on an intersection
of 2 lines, which are defined by 4 points. It also holds that
every four points on circumference define a point inside,
So there are n choose 4, (nC4) points inside. Together there
are V=n+nC4 points. Now let's count the edges. First we
will count the edges that we have drawn. We connect first
point with zero points, second point with 1 other, third point
with 2 others… We recognize the sum and determine
that there are n(n-1)/2 =nC2 connections. However every
inside point splits 2 lines and therefore generates 2 more
lines. That gives nC2+2*(number of inside points)=nC2+2(nC4), Adding the lines of circumference gives
us E=n+nC2+2(nC4).
We put all into the original equation to get F=2+E-V=2+n+nC2+2(nC4)-n-nC4=2+nC2+nC4. Since we
don't care about the region outside the circle, we get 1+nC2+nC4 which for n=10 gives us 256 parts.
2. If Miha and Gregor are both 10, and Katarina is 20. How old are Rok, Lucija and Ivo if you follow
this system of numbers?
3. If a pencil and a rubber cost 110c together, however a rubber costs 100c more than a pencil.
How much does the pencil cost?
4. You have 12 apples, one of them is heavier (odd) than the others. Let’s say that you can’t feel
by touch which one is heavier. Find out which apple is odd if you do exactly 3 weighings on a
balance scale.
Solution 2: Rok-5 , Lucija-15, Ivo-10 (5 for each syllable in a name)
Solution 3: 5c
Solution 3: You put 6 apples on both sides and keep the heavier side. You repeat the procedure with
3 apples on both sides and keep the heavier side. Finally you put one apple on both sides and keep
third one near you. If scale isn’t in balance, the odd apple is on the heavier side. If the scale is in
balance, apple near you is odd.
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