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4

Part of STEMS 2023 Math Cat A

Problems(2)

Sum of possible function values

Source: STEMS 2023 Maths CAT A Part A P4

1/8/2023
Let f:N→Nf : \mathbb{N} \to \mathbb{N} be a function such that the following conditions hold:
 (1)  f(1)=1.\qquad\ (1) \; f(1) = 1.
 (2)  (x+y)2<f(x+y)≤f(x)+f(y)  ∀  x,y∈N.\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.
 (3)  f(4n+1)<2f(2n+1)  ∀  n≥0.\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.
 (4)  f(4n+3)≤2f(2n+1)  ∀  n≥0.\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.
Find the sum of all possible values of f(2023)f(2023).
functionComputationalSTEMSalgebra
Alice and Bob playing with polynomials

Source: STEMS Mathematics 2023 Category B P2

1/8/2023
Alice has n>1n > 1 one variable quadratic polynomials written on paper she keeps secret from Bob. On each move, Bob announces a real number and Alice tells him the value of one of her polynomials at this number. Prove that there exists a constant C>0C > 0 such that after Cn5Cn^5 questions, Bob can determine one of Alice’s polynomials.
Proposed by Rohan Goyal and Anant Mudgal
algebrapolynomial