MathDB

P1

Part of Russian TST 2020

Problems(2)

Finances

Source: Russian TST 2020, Day 6 P1

3/21/2023
There are coins worth 1,2,…,b1, 2, \ldots , b rubles, blue bills with worth aa{} rubles and red bills worth a+ba + b rubles. Ilya wants to exchange a certain amount into coins and blue bills, and use no more than a−1a-1 coins. Pasha wants to exchange the same amount in coins and red bills, but use no more than aa{} coins. Prove that they have equally many ways of doing so.
combinatoricsnumber theory
Functional equation

Source: Russian TST 2020, Day 5 P1

3/21/2023
Determine all functions f:R+→R+f:\mathbb{R}^+\to\mathbb{R}^+ satisfying xf(xf(2y))=y+xyf(x)xf(xf(2y))=y+xyf(x) for all x,y>0x,y>0.
algebrafunctional equation