1 Number Theory AIME 2022 #13 · Intermediate Number Theory
AoPS 5.00 ▾

There is a polynomial $P(x)$ with integer coefficients such that\[P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\]holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$.

Working vocabulary 4
Coefficient Know what it means to find the coefficient of $x^{2022}$ in an expanded polynomial expression.
Exponentiation Be fluent manipulating expressions like $x^{2310}$ and $x^{105}$ using standard power rules.
Polynomial Know what a polynomial with integer coefficients is; be fluent reading and evaluating expressions in $x$.
Rational function Understand that the right-hand side is a ratio of two polynomial-like expressions that is claimed to equal a polynomial.
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2 Number Theory AIME 2016 #12 AI tagged · Intermediate Number Theory
AoPS 5.00 ▾

Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.

Working vocabulary 5
Divisibility check whether small primes divide a given integer.
Integer factorization comfortably factor an integer into its prime factors.
Positive integer know what positive integers are and how to iterate $m = 1, 2, 3, \dots$.
Prime number recognize primes and be able to test small numbers for primality.
Quadratic polynomial evaluate expressions of the form $m^2 - m + 11$ for given $m$.
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3 Counting & Probability AIME 2005 #5 · Intermediate Combinatorics
AoPS 4.00 ▾

Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

Working vocabulary 5
Combination be comfortable counting arrangements of indistinguishable objects in a line, like choosing $4$ of $8$ positions.
Multiplication principle combine independent counting choices by multiplying their counts.
Permutation know how to count orderings of a sequence and adjust for indistinguishable items.
Sequence be able to model a stack as an ordered list of $8$ items and reason about adjacency.
Symmetry recognize when two configurations are the "same" arrangement to avoid overcounting.
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4 Counting & Probability AMC12 2006 #25 · Intermediate Combinatorics
AoPS 4.00 ▾

How many non- empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties?

$(1)$ No two consecutive integers belong to $S$.

$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.

$\mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{(D) \ } 377\qquad \mathrm{(E) \ } 405$

Working vocabulary 5
Cardinality understand what it means for $S$ to contain $k$ elements.
Inequality (mathematics) parse the condition "no number less than $k$" as $\min(S) \geq k$.
Integer recognize consecutive integers like $n$ and $n+1$ within $\{1,\ldots,15\}$.
Set (mathematics) be comfortable reading subset notation and listing elements of $\{1,2,\ldots,15\}$.
Subset know what non-empty subsets are and how to count elements of a subset.
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5 Algebra AIME 2012 #2 AI tagged · Intermediate Algebra
AoPS 3.00 ▾

The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. Find the sum of the first, last, and middle terms of the original sequence.

Working vocabulary 5
Arithmetic progression be comfortable with the general term $a_k = a_1 + (k-1)d$ and how consecutive terms relate.
Linear equation solve basic linear equations in one or two unknowns from given numeric conditions.
Odd number recognize that the $k$th odd positive integer is $2k-1$.
Sequence read indexed notation $a_1, a_2, \ldots, a_n$ and identify first, last, and middle positions.
Summation know how to add a finite list of terms and manipulate sums like $\sum (a_k + b_k)$.
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6 Number Theory AIME 2019 #9 · Intermediate Number Theory
AoPS 4.50 ▾

Let $\tau(n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.

Working vocabulary 5
Divisor be fluent in finding all positive divisors of an integer by trial.
Divisor function know that $\tau(n)$ counts positive divisors of $n$ and how to compute it for small $n$.
Equation solving parse the equation $\tau(n)+\tau(n+1)=7$ and test candidate values of $n$.
Integer comfort with positive integers and basic arithmetic on them.
Prime factorization be able to factor small integers into primes quickly and reliably.
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