MathDB

10

Part of 2018 Hanoi Open Mathematics Competitions

Problems(4)

HOMC 2018

Source:

1/25/2019
Let T=14x2−15y2+16z2T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2} where x,y,zx,y,z are real numbers such that 1≤x,y,z≤41 \leq x,y,z \leq 4 and x−y+z=4x-y+z=4. Find the smallest value of 10×T10 \times T.
homchomc2018
problem of painting the central tower model (HOMC 2018 Team j10)

Source:

1/31/2020
[THE PROBLEM OF PAINTING THE THÁP RÙA (THE CENTRAL TOWER) MODEL] The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of 33 levels. For the first and second levels, each has 1010 doorways among which 33 doorways are located at the front, 33 at the back, 22 on the right side and 22 on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: 1. The top level is painted with only one color. 2. In the second level, the 33 doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors. 3. For the first level, we apply the same rules as for the second level. https://cdn.artofproblemsolving.com/attachments/2/3/18ee062b79693c4ccc26bf922a7f54e9f352ee.png (a) In how many ways the first level can be painted? (b) In how many ways the whole tower model can be painted?
combinatoricsColoring
100 students from 2 clubs A,B in circle (HOMC 2018 ind. sen10)

Source:

2/2/2020
There are 100100 school students from two clubs AA and BB standing in circle. Among them 6262 students stand next to at least one student from club AA, and 5454 students stand next to at least one student from club BB. 1) How many students stand side-by-side with one friend from club AA and one friend from club BB? 2) What is the number of students from club AA?
combinatoricscircle
problem of painting the central tower model (HOMC 2018 Team S10)

Source:

2/17/2020
The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of 33 levels. For the first and second levels, each has 1010 doorways among which 33 doorways are located at the front, 33 at the back, 22 on the right side and 22 on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level (Figure). We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: (a) The top level is painted with only one color. (b) The 33 doorways at the front on the second level are painted with the same color. (c) The 33 doorways at the front on the first level are painted with the same color. (d) Each of the remaining 1414 doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors. How many ways are there to paint the first level? How many ways are there to paint the entire tower model? https://cdn.artofproblemsolving.com/attachments/f/9/2249f8595a8efe711680f3dfb8ff959c140a21.png
Coloringcombinatorics