Gillis Mathematical Olympiad, 2019-2020
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Question 1 - Weighing Coins
Given are seven outwardly identical coins; four are genuine and three are counterfeit. The three counterfeit coins are of identical weight, as are the four genuine coins.
It is known that a counterfeit coin is lighter than a genuine coin. In one weighing, you can select two groups of coins and determine which is lighter, or if they have the same weight.
How many weighings are needed to locate at least one counterfeit coin? -
Question 2 - Mathematical Conference
202 participants from three countries attended a mathematical conference: Israel, Greece, and Japan.
On the first day, every pair of participants from the same country shook hands. On the second day, every pair of participants
where one was Israeli and the other was not Israeli shook hands. On the third day, every pair of participants where one
was Israeli and the other was Greek shook hands. In total, 20200 handshakes occurred. How many
Israeli participants were at the conference?Topics:Number Theory Combinatorics Algebra -> Word Problems Algebra -> Equations -> Diophantine Equations -
Question 3 - Where is the point?
In a convex hexagon ABCDEF, triangles ACE and BDF are congruent and regular. Show that the three segments connecting the midpoints of opposite sides of the hexagon intersect at one point.
Topics:Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Vectors -
Question 4 - Numbers on a Board
At the beginning of the day, four integers are written on the board (`a_0,b_0,c_0,d_0`). Every minute, Danny replaces the four numbers on the board with a new set of four numbers according to the following rule: If the numbers written on the board are (a,b,c,d), Danny first generates the numbers
`a'=a+4b+16c+64d`
`b'=b+4c+16d+64a`
`c'=c+4d+16a+64b`
`d'=d+4a+16b+64c`
Then he erases the numbers (a,b,c,d) and writes in their place the numbers (a',d',c',b'). For which initial sets (`a_0,b_0,c_0,d_0`) will Danny eventually write a set of four numbers that are all divisible by `5780^5780` -
Question 5 - Cyclic Quadrilaterals
Given two triangles ACE, BDF
intersecting at 6 points: G,H,I,J,K,L
as shown in the figure. It is given that in each of the quadrilaterals
EFGI, DELH, CDKG, BCJL, ABIK a circle can be inscribed.
Is it possible that a circle can also be inscribed in quadrilateral FAHJ?
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Question 6 - The Number Circle
The numbers 1 through 6 are written in a circle in order, as shown in the figure.
In each step, Lior chooses a number a in the circle whose neighbors are b and c, and replaces
it with the number `(bc)/a`.
Can Lior reach a state where the product of the numbers in the circle is greater than `10^100`
(a) in 100 steps.
(b) in 110 steps
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Question 7 - Inscribed Circle in a Triangle
Inside a triangle there is a point P, whose distances from the lines containing the sides of the triangle are `d_a,d_b,d_c`. Let R denote the radius of the circumscribed circle of the triangle and r the radius of the inscribed circle in the triangle. Show that `sqrt(d_a)+sqrt(d_b)+sqrt(d_3)<= sqrt (2R+5r) `.
Topics:Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Circles Algebra -> Inequalities