MathDB

2020 Durer Math Competition Finals

Part of Durer Math Competition

Subcontests

(16)
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6 VIP chairs in a movie theatre

In a movie theatre there are 66 VIP chairs labelled from 11 to 66. We call a few consecutive vacant chairs a block. In the online VIP seat reservation process the reservation of a seat consists of two steps: in the first step we choose the block, in the second step we reserve the first, last or middle seat (in case of a block of size even this means the middle chair with the smaller number) of that block. (In the second step the online system offers the three possibilities even though they might mean the same seat.) Benedek reserved all seats at some screeining. In how many ways could he do it if we distinguish two reservation if there were a step when Benedek chose a different option?
For instance, if the seats 1 1 and 66 are reserved, then there are two blocks, the first one consists of the seat 1 1, the second block consists of the seats 3,43, 4 and 55. Two reservation orders are different if there is a chair that was reserved in a different step, or there is a chair that was reserved with different option (first, last or middle). So if there were 22 VIP chairs, then the answer would have been 99.

f(n) = (p_1-1)^{k_1+1}(p_2-1)^{k_2+1}...(pt-1)^{k_t+1}

The function ff is defined on positive integers : if nn has prime factorization p1k1p2k2...ptktp^{k_1}_{1} p^{k_2}_{2} ...p^{k_t}_{t} then f(n)=(p1−1)k1+1(p2−1)k2+1...(pt−1)kt+1f(n) = (p_1-1)^{k_1+1}(p_2-1)^{k_2+1}...(p_t-1)^{k_t+1}. If we keep using this function repeatedly, starting from any positive integer nn, we will always get to 11 after some number of steps. What is the smallest integer nn for which we need exactly 66 steps to get to 11?
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square inscribed in triangle, sum of reciprocals of 3 inradii

In triangle ABCABC we inscribe a square such that one of the sides of the square lies on the side ACAC, and the other two vertices lie on sides ABAB and BCBC. Furthermore we know that AC=5AC = 5, BC=4BC = 4 and AB=3AB = 3. This square cuts out three smaller triangles from △ABC\vartriangle ABC. Express the sum of reciprocals of the inradii of these three small triangles as a fraction p/qp/q in lowest terms (i.e. with pp and qq coprime). What is p+qp + q?

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

In the game of Yahtzee , players have to achieve various combinations of values with 55 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 66’s in a round, given that he plays with the optimal strategy to maximise this probability?
Writing the answer as p/qp/q where pp and qq are coprime, you should submit the sum of all prime factors of pp, counted with multiplicity. So for example if you obtained pq=34⋅1125⋅5\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5} then the submitted answer should be 4⋅3+11=234 \cdot 3 + 11 = 23.
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Santa Claus game, present behind 1 of 100 doors

Santa Claus plays a guessing game with Marvin before giving him his present. He hides the present behind one of 100100 doors, numbered from 11 to 100100. Marvin can point at a door, and then Santa Claus will reply with one of the following words:
∙\bullet "hot" if the present lies behind the guessed door,
∙\bullet "warm" if the guess is not exact but the number of the guessed door differs from that of the present’s door by at most 55,
∙\bullet "cold" if the numbers of the two doors differ by more than 55.
At least how many such guesses does Marvin need, so that he can be certain about where his present is?
Marvin does not necessarily need to make a "hot" guess, just to know the correct door with 100%100\% certainty.

red and blue balls in an urn , 1024 in total

There are red and blue balls in an urn : 10241024 in total. In one round, we do the following: we draw the balls from the urn two by two. After all balls have been drawn, we put a new ball back into the urn for each pair of drawn balls: the colour of the new ball depends on that of the drawn pair. For two red balls drawn, we put back a red ball. For two blue balls, we put back a blue ball. For a red and a blue ball, we put back a black ball. For a red and a black ball, we put back a red ball. For a blue and a black ball, we put back a blue ball. Finally, for two black balls we put back a black ball. Then the next round begins. After 1010 rounds, a single ball remains in the urn, which is red. What is the maximal number of blue balls that might have been in the urn at the very beginning?
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n integers on a paper, sum of their 2^n subsets on the blackboard

Endre wrote nn (not necessarily distinct) integers on a paper. Then for each of the 2n2^n subsets, Kelemen wrote their sum on the blackboard. a) For which values of nn is it possible that two different nn-tuples give the same numbers on the blackboard? b) Prove that if Endre only wrote positive integers on the paper and Ferenc only sees the numbers on the blackboard, then he can determine which integers are on the paper.

sum of digits of 3n if sum of digits of n is 100 and of 4n is 800

We have a positive integer nn, whose sum of digits is 100100 . If the sum of digits of 44n44n is 800800 then what is the sum of digits of 3n3n?

collinear wanted, perpend. to angle bisector passing through incenter

Let ABCABC be a scalene triangle and its incentre II. Denote by FAF_A the intersection of the line BCBC and the perpendicular to the angle bisector at AA through II. Let us define points FBF_B and FCF_C in a similar manner. Prove that points FA,FBF_A, F_B and FCF_C are collinear.
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tank of enemy at one of the 13 x 13 fields

We are given a map divided into 13×1313\times 13 fields. It is also known that at one of the fields a tank of the enemy is stationed, which we must destroy. To achieve this we need to hit it twice with shots aimed at the centre of some field. When the tank gets hit it gets moved to a neighbouring field out of precaution. At least how many shots must we fire, so that the tank gets destroyed certainly? We can neither see the tank, nor get any other feedback regarding its position.

equal in different bases 11001_? = 54001_{10}

What number should we put in place of the question mark such that the following statement becomes true? 11001?=540011011001_? = 54001_{10} A number written in the subscript means which base the number is in.
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ATOC and BTOC have equal area, circumcenter, altitude

Let ABCABC be an acute triangle where AC>BCAC > BC. Let TT denote the foot of the altitude from vertex CC, denote the circumcentre of the triangle by OO. Show that quadrilaterals ATOCATOC and BTOCBTOC have equal area.

dominoes in a 3x3 square

How many ways are there to tile a 3×33 \times 3 square with 44 dominoes of size 1×21 \times 2 and 11 domino of size 1×11 \times 1?
Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical

Help! USAMO 1991 #3

Show that, for any fixed integer  n≥1, \,n \geq 1,\, the sequence 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) is eventually constant.
[The tower of exponents is defined by a1=2,  ai+1=2aia_1 = 2, \; a_{i+1} = 2^{a_i}. Also a_i \; (\mbox{mod} \; n) means the remainder which results from dividing aia_i by nn.]