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2023 SSMO Relay Round 2 Problems

Revision as of 20:33, 15 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem 1== Consider the cubic polynomial <math>P(x)=ax^3+bx^2+cx+d</math>, where <math>a,b,c,d</math> are single-digit integers, which has roots of approximately <cmath>x...")
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Problem 1

Consider the cubic polynomial $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are single-digit integers, which has roots of approximately \[x \approx -0.9518399, 0.2055095, 1.460616.\] Compute $|f(3)|$.

Solution

Problem 2

Let $T=$ TNYWR. Suppose that $L = \left\lfloor\sqrt{N}\right\rfloor$ points are evenly spaced around the circle. Find the number of ways to select $k \ge 3$ points such that the $k$-gon formed strictly contains the center of the circle.

Solution

Problem 3

Let $T=$ TNYWR. In a committee of $2023$ people, $N$ are scientists and the rest are builders. In order to make a building, $\frac{N}{2}$ people must be choosen with at least one scientist and one builder. If $x$ is the number of ways to do this, find the largest integer $a$ such $2^a \mid x$.

Solution

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