13

Part of 2020 Durer Math Competition Finals

Problems(2)

square inscribed in triangle, sum of reciprocals of 3 inradii

Source: 2020 Dürer Math Competition Finals Day2 E13 https://artofproblemsolving.com/community/c1622639_2020_

ABC

we inscribe a square such that one of the sides of the square lies on the side

AC

, and the other two vertices lie on sides

AB

BC

. Furthermore we know that

AC = 5

BC = 4

AB = 3

. This square cuts out three smaller triangles from

\vartriangle ABC

. Express the sum of reciprocals of the inradii of these three small triangles as a fraction

p/q

in lowest terms (i.e. with

p

q

coprime). What is

p + q

squaregeometryinscribedinradius

5 dice, probability that a player achieves at least four 6’s in a round

Source: 2020 Dürer Math Competition Finals Day2 E+13 https://artofproblemsolving.com/community/c1622639_2020_

In the game of Yahtzee , players have to achieve various combinations of values with

5

dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four

6

’s in a round, given that he plays with the optimal strategy to maximise this probability?
Writing the answer as

p/q

p

q

are coprime, you should submit the sum of all prime factors of

p

, counted with multiplicity. So for example if you obtained

\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}

then the submitted answer should be

4 \cdot 3 + 11 = 23

probabilitycombinatorics