MathDB

10

Part of 2023 MOAA

Problems(4)

2023 MOAA Speed P10

Source:

10/13/2023
If x,y,zx,y,z satisfy the system of equations xy+yz+zx=23xy+yz+zx=23 yx+y+zy+z+xz+x=−1\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1 z2xx+y+x2yy+z+y2zz+x=202\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202 Find the value of x2+y2+z2x^2+y^2+z^2.
Proposed by Harry Kim
MOAA 2023
2023 MOAA Accuracy P10

Source:

10/13/2023
Let SS be a set of integers such that if aa and bb are in SS then 3a−2b3a-2b is also in SS. How many ways are there to construct SS such that SS contains exactly 44 elements in the interval [0,40][0,40]?
Proposed by Harry Kim
MOAA 2023
2023 MOAA Team P10

Source:

10/14/2023
Let SS be the set of lattice points (a,b)(a,b) in the coordinate plane such that 1≤a≤301\le a\le 30 and 1≤b≤301\le b\le 30. What is the maximum number of lattice points in SS such that no four points form a square of side length 2?
Proposed by Harry Kim
MOAA 2023
2023 MOAA Gunga P10

Source:

10/15/2023
A number is called winning if it can be expressed in the form a20+b23\frac{a}{20}+\frac{b}{23} where aa and bb are positive integers. How many winning numbers are less than 1?
Proposed by Andy Xu
MOAA 2023