MathDB

P2

Part of 2021/2022 Tournament of Towns

Problems(5)

Moving shapes to form polygon

Source: 43rd International Tournament of Towns, Junior A-Level P2, Fall 2021

2/18/2023
On a table there are all 8 possible horizontal bars 1×31\times3 such that each 1×11\times1 square is either white or gray (see the figure). It is allowed to move them in any direction by any (not necessarily integer) distance. We may not rotate them or turn them over. Is it possible to move the bars so that they do not overlap, all the white points form a polygon bounded by a closed non-self-intersecting broken line and the same is true for all the gray points?
Mikhail Ilyinsky
combinatoricsTournament of Towns
Cube split into parallelepipeds

Source: 43rd International Tournament of Towns, Junior O-Level P2 & Senior O-Level P2, Fall 2021

2/18/2023
A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds.
Oleg Smirnov
geometryTournament of Towns
2022 TT Spring SA2

Source: 2022 Tournament of towns Spring Senior A-level

5/11/2022
On a blank paper there were drawn two perpendicular axes xx and yy with the same scale. The graph of a function y=f(x)y=f(x) was drawn in this coordinate system. Then the yy axis and all the scale marks on the xx axis were erased. Provide a way how to draw again the yy axis using pencil, ruler and compass: (a) f(x)=3xf(x)= 3^x; (b) f(x)=log⁡axf(x)= \log_a x, where a>1a>1 is an unknown number.
functionanalytic geometrygeometryRulercompass
2022 TT spring SO2

Source: 2022 Tournament of towns Spring Senior O level

5/11/2022
The fox and pinocchio have grown a tree on the field of miracles with 88 golden coins. It is known that exactly 33 of them are counterfeit. All the real coins weigh the same, the counterfeit coins also weigh the same but are lighter. The fox and pinocchio have collected the coins and wish to divide them. The fox is going to give 3 coins to pinocchio, but pinocchio wants to check whether they all are real. Can he check this using 22 weighings on a balance scale with no weights?
combinatorics
Tournament of Towns O-level Question 2

Source:

5/14/2022
Peter picked an arbitrary positive integer, multiplied it by 5, multiplied the result by 5, then multiplied the result by 5 again and so on. Is it true that from some moment all the numbers that Peter obtains contain 5 in their decimal representation?
number theory