Difference between revisions of "AIME 2020(MOCK) Problems"

Latest revision as of 05:45, 23 December 2020

Problem 1

Let $N$ be $112123123412345... (1000 digits)$ . What is the remainder when $N$ is divided by $21$ ?

Problem 2

Let $K$ be a set of polynomials $P(x)$ with integral coefficients such that the roots of $P(x)$ are $cos \frac{\pi}{7}$ , $cos \frac{\pi}{11}$ , and $cos \frac{\pi}{17}$ . What is the least possible sum of the coefficients of $P(x)$ ?

Problem 3

How many $15$ digit base $5$ positive integers consist of exactly $2$ pairs of consecutive $0$ s but no $4$ consecutive $3$ s?

Problem 4

Let $\lfloor\x\rfloor$ (Error compiling LaTeX. Unknown error_msg) denote the greatest integer less than or equal to $x$ . What is the tens digit of $\lfloor\frac{10^{2020}}{10^{101} + 7}\rfloor$

@@ Line 6: / Line 6: @@
 ==Problem 2==
-Let <math>K</math> be a set of polynomials <math>P(x)</math> such that the roots of <math>P(x)</math> are <math>cos \frac{\pi}{2}</math>, <math>cos \frac{\pi}{4}</math>
+Let <math>K</math> be a set of polynomials <math>P(x)</math> with integral coefficients such that the roots of <math>P(x)</math> are <math>cos \frac{\pi}{7}</math>, <math>cos \frac{\pi}{11}</math>, and <math>cos \frac{\pi}{17}</math>. What is the least possible sum of the coefficients of <math>P(x)</math>?
+==Problem 3==
+How many <math>15</math> digit base <math>5</math> positive integers consist of exactly <math>2</math> pairs of consecutive <math>0</math>s but no <math>4</math> consecutive <math>3</math>s?
+==Problem 4==
+Let <math>\lfloor\x\rfloor</math> denote the greatest integer less than or equal to <math>x</math>. What is the tens digit of <math>\lfloor\frac{10^{2020}}{10^{101} + 7}\rfloor</math>

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