Olympiad problems

2020 AIME Problems, 11

Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2020 AIME Problems, 1

Tags: AOIME , 2020 AOIME

Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.

2020 AIME Problems, 3

Tags: AOIME , 2020 AOIME , logarithms

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2020 AIME Problems, 13

Tags: 2020 AOIME , 2020 AIME II

Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.

2020 AIME Problems, 4

Tags: geometry , geometric transformation , rotation , AOIME , 2020 AOIME

Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.

2020 AIME Problems, 9

Tags: 2020 AOIME , 2020 AIME II

While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.

2020 AIME Problems, 14

Tags: 2020 AOIME , algebra

For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\leq x\leq 2020$. Find the remainder when $N$ is divided by $1000$.