MathDB

Problem 4

Part of 2001 Federal Math Competition of S&M

Problems(3)

removing coins from a pile

Source: Serbia & Montenegro 2001 1st Grade P4

6/1/2021
There are nn coins in the pile. Two players play a game by alternately performing a move. A move consists of taking 5,75,7 or 1111 coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if:
(a) n=2001n=2001; (b) n=5000n=5000?
combinatoricsgame
arithmetic means given, find max/min(max)

Source: Serbia & Montenegro 2001 2nd Grade P4

6/2/2021
Let SS be the set of all nn-tuples of real numbers, with the property that among the numbers x1,x1+x22,…,x1+x2+…+xnnx_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n the least is equal to 00, and the greatest is equal to 11. Determine max⁡(x1,x2,…,xn)∈Smax⁡1≤i,j≤n(xi−xj)andmin⁡(x1,x2,…,xn)∈Smax⁡1≤i,j≤n(xi−xj).\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).
inequalitiesalgebra
pyramid, parallelogram base

Source: Serbia & Montenegro 2001 3,4th Grade P4

6/2/2021
Parallelogram ABCDABCD is the base of a pyramid SABCDSABCD. Planes determined by triangles ASCASC and BSDBSD are mutually perpendicular. Find the area of the side ASDASD, if areas of sides ASB,BSCASB,BSC and CSDCSD are equal to x,yx,y and zz, respectively.
geometry3D geometrypyramidparallelogram