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Part of 2021 South East Mathematical Olympiad

Problems(2)

China South East Mathematical Olympiad 2021 Grade10 P8

Source:

7/30/2021
Determine all the pairs of positive integers (a,b),(a,b), such that 14φ2(a)−φ(ab)+22φ2(b)=a2+b2,14\varphi^2(a)-\varphi(ab)+22\varphi^2(b)=a^2+b^2, where φ(n)\varphi(n) is Euler's totient function.
number theoryEuler s totient functionBashing
China South East Mathematical Olympiad 2021 Grade11 P8

Source:

8/8/2021
A sequence {zn}\{z_n\} satisfies that for any positive integer i,i, zi∈{0,1,⋯ ,9}z_i\in\{0,1,\cdots,9\} and zi≡i−1(mod10).z_i\equiv i-1 \pmod {10}. Suppose there is 20212021 non-negative reals x1,x2,⋯ ,x2021x_1,x_2,\cdots,x_{2021} such that for k=1,2,⋯ ,2021,k=1,2,\cdots,2021, ∑i=1kxi≥∑i=1kzi,∑i=1kxi≤∑i=1kzi+∑j=11010−j50zk+j.\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}. Determine the least possible value of ∑i=12021xi2.\sum_{i=1}^{2021}x_i^2.
minimum valuealgebra