Olympiad problems

2018 AIME Problems, 1

Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.

2018 AIME Problems, 6

Tags: complex numbers , AMC , AIME , AIME I , 2018 AIME I

Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.

2018 AIME Problems, 11

Tags: AIME I , AMC , AIME , 2018 AIME I , bash , Lifting the Exponent

Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.

2018 AIME Problems, 9

Tags: 2018 AIME I , combinatorics

Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets.

2018 AIME Problems, 14

Tags: 2018 AIME I , AMC , AIME , AIME I

Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.

2018 AIME Problems, 2

Tags: AMC , AIME , AIME I , 2018 AIME I

The number $n$ can be written in base $14$ as $\underline{a}$ $\underline{b}$ $\underline{c}$, can be written in base $15$ as $\underline{a}$ $\underline{c}$ $\underline{b}$, and can be written in base $6$ as $\underline{a}$ $\underline{c}$ $\underline{a}$ $\underline{c}$, where $a > 0$. Find the base-$10$ representation of $n$.