MathDB

2

Part of 2021 Israel TST

Problems(5)

Switches randomly connected to bulbs.

Source: ISR 2021 TST1 p.2

5/4/2022
Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch
combinatorics
Bigger than 1.

Source: ISR 2021 TST2 p.2

5/4/2022
Suppose x,y,z∈R+x,y,z\in \mathbb R^+. Prove that xyz+4xy+4xz+yzx+4yz+4yx+zxy+4zx+4zy≥1\frac {x}{\sqrt{yz+4xy+4xz}}+\frac {y}{\sqrt{zx+4yz+4yx}}+\frac {z}{\sqrt{xy+4zx+4zy}}\geq 1.
inequalities
Unbounded functions.

Source: ISR 2021 February TST p.2

5/4/2022
Find all unbounded functions f:Z→Zf:\mathbb Z \rightarrow \mathbb Z , such that f(f(x)−y)∣x−f(y)f(f(x)-y)|x-f(y) holds for any integers x,yx,y.
functional equationfunctional equation in Zalgebra
Game in Euclidian plane, distance average

Source: 2021 Israel TST Test 6 P2

7/25/2022
Let n>1n>1 be an integer. Hippo chooses a list of nn points in the plane P1,…,PnP_1, \dots, P_n; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list XX and measures the distances from it to the other n−1n-1 points in the list. The average of the resulting n−1n-1 numbers will be denoted m(X)m(X). Find all values of nn for which Hippo can prepare the list in such a way, that for any point XX Wombat may pick, he can point to a point YY from the list such that XY=m(X)XY=m(X).
combinatorial geometrydistancesTSTcombinatorics
Variant on old FE, f(xf(x+y))

Source: 2021 Israel TST 8 P2

5/31/2022
Find all functions f:R→Rf:\mathbb{R}\to \mathbb{R} so that for any reals x,yx,y the following holds: f(x⋅f(x+y))+f(f(y)⋅f(x+y))=(x+y)2f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2