MathDB

2021 Durer Math Competition Finals

Part of Durer Math Competition

Subcontests

(16)
12
1
16
2

remove some internal walls in a 2x7 squaretable

Consider a table consisting of 2×72\times 7 squares. Each little square is surrounded by walls (each internal wall belongs to two squares). We would like to remove some internal walls to make it possible to get from any square to any other one without crossing walls. How many ways can we do this while removing the minimal possible number of internal walls? The figure shows a possible configuration, the remaining walls are marked in red, the removed ones are marked in light pink. Two configurations are considered the same if the same walls are removed. https://cdn.artofproblemsolving.com/attachments/d/c/1a3d9ab0d0971929e6d484a970d4b1f36f0031.png

convex quadrilateral with angles and sides in 2 arithmetic sequences

The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length 11, then its perimeter is a+b4a + \sqrt{b}4, where a a and bb are positive integers. What is the value of a+ba + b?
10
2

area of triangle, medians

A triangle is given. Its side a is of length 2020 cm, and its area is 125125 cm2^2. It is also known that one of the angles lying on side a is twice as large as the other one. We cut the triangle into two parts at the median belonging to side a. Then we move the so-obtained two parts towards each other, such that the two segments of side a remain on the same line (i.e., the line initially occupied by side a). We move the two parts towards each other until we first reach a moment when the common part of the two segments is of length 44 cm. What is the area of the so-obtained shape in cm2^2? The so-obtained shape is the union of the two parts, which is a heptagon. https://cdn.artofproblemsolving.com/attachments/3/0/3d45e2df6a0043dfa4fe5ccf64865da8879b42.png

max no r of horses Billy can own standing on lattice points

Billy owns some really energetic horses. They are hopping around on points of the plane having two integer coordinates. Each horse can make the following types of jumps. They can hop to a point obtained from their current position via a translation by vector (15,9)(15, 9), (−9,15)(-9, 15), (−15,−9)(-15,-9) or (9,−15)(9,-15). The horses are now standing on lattice points such that no two can meet by making jumps as above. What is the maximal possible number of horses Billy can own?
15
1
14
1
13
2
11
1

American makes a living from trading cows

Japanese businessman Rui lives in America and makes a living from trading cows. On Black Thursday he was selling his cows for 20002000 dollars each (the cows were of the same price), but after the financial crash there were huge fluctuations in the market and Rui was forced to follow them with his pricing. Every day he doubled, halved, multiplied by five or divided by five the price from the previous day (even if it meant he had to give change in cents). At the same time he managed to follow the Japanese superstition, so that the integer part of the price in dollars never started with digit 44. On the day when Billy visited him to buy some cows the price of each cow was 8080 dollars. What is the minimal number of days that could have passed since Black Thursday by then?
9
1
8
2

g(x) = (x -1)^2(p + 2x)^2

John found all real numbers pp such that in the polynomial g(x)=(x−1)2(p+2x)2g(x) = (x -1)^2(p + 2x)^2 , the quadratic term has coefficient 20212021. What is the sum of all of these values pp?

no of true statements about 300 concerning disibility

Benedek wrote the following 300300 statements on a piece of paper. 2∣1!2 | 1! 3∣1!   3∣2!3 | 1! \,\,\, 3 | 2! 4∣1!   4∣2!   4∣3!4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3! 5∣1!   5∣2!   5∣3!   5∣4!5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4! ...... 24∣1!   24∣2!   24∣3!   24∣4!   ⋅⋅⋅   24∣23!24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23! 25∣1!   25∣2!   25∣3!   25∣4!   ⋅⋅⋅   25∣23!   25∣24!25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24! How many true statements did Benedek write down? The symbol | denotes divisibility, e.g. 6∣4!6 | 4! means that 66 is a divisor of number 4!4!.
7
1
2
3

least sum of properties of all cities

In the country of Óxisz the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring city counted in dinar is divisible by 10001000, and she also observed that the sum of properties of all cities is also divisible by 10001000. What is the least sum of properties of all cities if the map of the cities looks as follows? https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png Remark: The cities may have non-integer properties, but it is also positive. On the map the points are the cities, and two cities are neighboring if there is a direct connection between them.

2^n+2^{n+3} is a perfect square

Find the number of integers nn between 11 and 20212021 such that 2n+2n+32^n+2^{n+3} is a perfect square.

in a french village the number of inhabitants is a perfect square

In a french village the number of inhabitants is a perfect square. If 100100 more people moved in, then the number of people would be 1 1 bigger than a perfect square. If again 100100 more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?
6
2

game with 15 totem poles in an Indian reservatory

(Game) In an Indian reservatory there are 1515 totem poles arranged according to the left figure. Silent Stream and Red Fire used to play the following game: In turns they stretch ropes between two-two poles in such a way that every stretched rope is parallel to a side of the big triangle and no rope can go along a pole that is already touched by another rope. Furthermore, if instead of a rope one can stretch out a straight line extension of the rope, then one should stretch out this extension. The one who cannot stretch out more rope according to the rules loses. Win two games in a row against the organizers! You can decide that you want to start or to be the second player. The figure on the right depicts the first three steps of a game. First Silent Stream stretches the blue rope, then Red Fire stretches the red one, finally Silent Stream stretches the blue one. https://cdn.artofproblemsolving.com/attachments/f/8/3b8b9e38a8a477da288566ecb26036bfc7e615.png

4-digit positive number. and simple graph on 4 vertices

Bertalan thought about a 44-digit positive number. Then he draw a simple graph on 44 vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.
5
4
4
2
3
3

5 points concyclic wanted

Let AA and BB different points of a circle kk centered at OO in such a way such that ABAB is not a diagonal of kk. Furthermore, let XX be an arbitrary inner point of the segment ABAB. Let k1k_1 be the circle that passes through the points AA and XX, and AA is the only common point of kk and k1k_1. Similarly, let k2k_2 be the circle that passes through the points BB and XX, and BB is the only common point of kk and k2k_2. Let MM be the second intersection point of k1k_1 and k2k_2. Let QQ denote the center of circumscribed circle of the triangle AOBAOB. Let O1O_1 and O2O_2 be the centers of k1k_1 and k2k_2. Show that the points M,O,O1,O2,QM,O,O_1,O_2,Q are on a circle.

a line intersecting a square lattice

The figure shows a line intersecting a square lattice. The area of some arising quadrilaterals are also indicated. What is the area of the region with the question mark? https://cdn.artofproblemsolving.com/attachments/0/d/4d5741a63d052e3f6971f87e60ca7df7302fb0.png

n kids with k bars of chocolate

On the evening of Halloween a group of nn kids collected kk bars of chocolate of the same type. At the end of the evening they wanted to divide the bars so that everybody gets the same amount of chocolate, and none of the bars is broken into more than two pieces. For which nn and kk is this possible?
1
3

prime = a^3-b^3 => p=6k+1

Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder 11 after division by 66.

in the k-th month the weeks consist of 6^{k-1} days

In Sixcountry there are 12 12 months, but each month consists of 66 weeks. The month are named the same way we do, from January to December, but in each month the weeks have different lengths. In the kk-th month the weeks consist of 6k−16^{k-1} days. What is the number of days of the spring (March, April, May together)?

computational geo, right triangle, another right angle

Given a right angled triangle ABCABC in which ∠ACB=90o\angle ACB = 90^o. Let DD be an inner point of ABAB, and let EE be an inner point of ACAC. It is known that ∠ADE=90o\angle ADE = 90^o, and that the length of the segment ADAD is 88, the length of the segment DEDE is 1515, and the length of segment CECE is 33. What is the area of triangle ABCABC?