MathDB

2022 Durer Math Competition Finals

Part of Durer Math Competition

Subcontests

(16)
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6 rows of doors, both rows are made up of three doors

In Durer’s duck school, there are six rows of doors, as seen on the diagram; both rows are made up of three doors. Dodo duck wishes to enter the school from the street in a way that she uses all six doors exactly once. (On her path, she may go to the street again, or leave the school, so long as she finishes her path in the school.) How many ways can she perform this? Two paths are considered different if Dodo takes the doors in a different order. https://cdn.artofproblemsolving.com/attachments/5/1/8b722eb2c642e8275928753921fdfbd7495df9.png

1^1+2^2+3^3+. . .+2021^{2021} mod 35

Benedek scripted a program which calculated the following sum: 11+22+33+...+202120211^1+2^2+3^3+. . .+2021^{2021}. What is the remainder when the sum is divided by 3535?
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numbers in 6x6 board

Write some positive integers in the following table such that ⋅\cdot there is at most one number in each field ⋅\cdot each number is equal to how many numbers there are in edge-adjacent fields, ⋅\cdot edge-adjacent fields cannot have equal numbers. What is the sum of numbers in the resulting table? https://cdn.artofproblemsolving.com/attachments/a/9/63a9c38762a4c895688fff049ed08c96b2c22c.png

computational with tangent circles

Circle k1k_1 has radius 1010, externally touching circle k2k_2 with radius 1818. Circle k3k_3 touches both circles, as well as the line ee determined by their centres. Let k4k_4 be the circle touching k2k_2 and k3k_3 externally (other than k1k_1) whose center lies on line ee. What is the radius of k4k_4?
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Doofy duck buy tangerines in the store

Doofy duck buy tangerines in the store. All tangerines have equal weight and are divided into 99, 1010, 1111, 1212, or 1313 equal wedges, although this cannot be seen without peeling them. How many tangerines does Doofy duck need to buy if he wishes to eat exactly one tangerine’s worth while eating at most one wedge from every tangerine? Doofy duck only peels the tangerines at home.

ant crawls along the grid lines of an infinite quadrille notebook

An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability 13\frac13 , left with probability 13\frac13 , and right with probability 13\frac13. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point? If the answer is pq\frac{p}{q} , where pp and qq are coprime positive integers, then your answer should be p+qp + q. The steps of the ant are independent.
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divide divisors into two groups with equal product if their product is a perfect

Show that the divisors of a number n≥2n \ge 2 can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of nn is a square number.

min no of regular triangles to cover the lines of a diagram

At least how many regular triangles are needed to cover the lines of the following diagram? https://cdn.artofproblemsolving.com/attachments/e/3/4de2ed2c7cc9421d7d060f0bc537ccaa3838fc.png (Only the perimeter of the triangles is involved in the covering, and the entire perimeter need not be incident on the diagram.)

GJ , FB , HK concurrent or parallel , ABCD cyclic and orthodiagonal

ABCDABCD is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let OO denote the centre of its circumcircle and EE the intersection of the diagonals. JJ and KK denote the perpendicular projections of EE on the sides ABAB and BCBC . Let F,GF , G and HH be the midpoint line segments. Show that lines GJGJ , FBFB and HKHK either pass through the same point or are parallel to each other.
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n numbered students , n pieces of paper

nn students, numbered from 11 to nn are sitting next to each other in a class. In the beginning the 11st student has nn pieces of paper in one pile. The goal of the students is to distribute the nn pieces in a way that everyone gets exactly one. The teacher claps once in a minute and for each clap the students can choose one of the following moves (or do nothing): ∙\bullet They divide one of their piles of paper into two smaller piles. ∙\bullet They give one of their piles of paper to the student with the next number. At least how many times does the teacher need to clap in order to make it possible for the students to distribute all the pieces of paper amongst themselves?

3 palaces, each rotating on a duck leg

Three palaces, each rotating on a duck leg, make a full round in 3030, 5050, and 7070 days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?

sum (x - yz)^2/(x + yz)^2 <1 if x,y,z>0 with x+y+z = 1, x> yz, y > zx, z > xy

Let x,y,zx, y, z denote positive real numbers for which x+y+z=1x+y+z = 1 and x>yzx > yz, y>zxy > zx, z>xyz > xy. Prove that (x−yzx+yz)2+(y−zxy+zx)2+(z−xyz+xy)2<1.\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.
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drawing X on 5x5 board

Anett is drawing XX-es on a 5×55 \times 5 grid. For each newly drawn XX she gets points in the following way: She checks how many XX-es there are in the same row (including the new one) that can be reached from the newly drawn XX with horizontal steps, moving only on fields that were previously marked with XX-es. For the vertical XX-es, she gets points the same way. a) What is the maximum number of points that she can get with drawing 2525 XX-es? b) What is the minimum number of points that she can get with drawing 2525 XX-es? For example, if Anett put the XX on the field that is marked with the circle, she would get 33 points for the horizontal fields and 11 point for the vertical ones. Thus, she would get 44 points in total. https://cdn.artofproblemsolving.com/attachments/5/c/662c2e4c3dea8d78e2f6397489b277daee0ad0.png

Csaba stands in the middle of a 15 x 15 m room

Csaba stands in the middle of a 1515 m ×15\times 15 m room at a workplace where everyone strictly adheres to 1,51,5 m social distancing. At least how many people are there other than Csaba in the room if Csaba cannot reach any wall without the others moving? The people are viewed as points.
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