LiveAoPSBench

Determine the number of permutations $a_1, a_2, a_3, \ldots, a_{109}$ of the integers $2, 3, 4, \ldots, 110$ such that for each $1 \leq k \leq 109$, the number $a_k$ is divisible by $k$.

Evaluate the definite integral:

$I = \int _0^1\frac{1}{\sqrt{1-x^2}}\ln \left(\frac{\sqrt{1+x}+1}{\sqrt{1+x}-1}\right)dx.$

Determine all polynomials $p(x)$ that satisfy the equation $(x-16)p(2x) = 16(x-1)p(x)$ for all $x$.

Find all integer solutions $n$ to the equation $\lfloor{\frac{n^2}{2}}\rfloor = \lfloor\frac{n}{2}\rfloor^{2}$.

For the sequence $\{a_n\}_{n \geq 1}$, let $a_1 = 2$ and for all integers $n \geq 1$, [ \left( a_{n+1} - \frac{3a_n + 1}{a_n + 3} \right) \left( a_{n+1} - \frac{5a_n + 1}{a_n + 5} \right) = 0. ] Determine the number of different values that $a_{2024}$ can take.

$\textbf{a)}\ 2024 \qquad\textbf{b)}\ 2024^2 \qquad\textbf{c)}\ 2^{2023} \qquad\textbf{d)}\ 2^{2024} \qquad\textbf{e)}\ \text{None}$

Determine all continuous functions $f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ that satisfy the inequality: $\frac{f(x)}{y^4} - \frac{f(y)}{x^4} \leq \left(\frac{1}{x} - \frac{1}{y}\right)^4$ for all $x, y > 0$.

Evaluate the limit without using derivatives: $\lim_{x \to 1}\dfrac{\frac{n}{1-x^n}-\frac{1}{1-x}-\frac{n-1}{2}}{x-1}$ where $n \in \mathbb{N_+}$.

In a convex quadrilateral $PQRS$, the side lengths are given as $PQ = 40$, $PS = 60$, and $RS = 20$. Given that $\angle QPS = \angle RSP = 60^\circ$, determine the measure of $\angle QRS$.

Inside a large quadrilateral, 10 small congruent squares, each with a side length of $1$ cm, are placed as shown below. Determine the area of the large quadrilateral in square centimeters. [asy] usepackage("tikz"); label(" \begin{tikzpicture}[scale=.7] \fill[white] (1,-1)--(4.5,2.5)--(1,6)--(-2.5,2.5)--cycle; \draw[thick] (1,-1)--(4.5,2.5)--(1,6)--(-2.5,2.5)--cycle; \fill[gray!80] (0,0) rectangle (1,1); \fill[gray!80] (1,0) rectangle (2,1); \fill[gray!80] (-1,1) rectangle (0,2); \fill[gray!80] (-2,2) rectangle (-1,3); \fill[gray!80] (2,1) rectangle (3,2); \fill[gray!80] (3,2) rectangle (4,3); \fill[gray!80] (-1,3) rectangle (0,4); \fill[gray!80] (2,3) rectangle (3,4); \fill[gray!80] (0,4) rectangle (1,5); \fill[gray!80] (1,4) rectangle (2,5); %== \draw[thick] (0,0) rectangle (1,1); \draw[thick] (1,0) rectangle (2,1); \draw[thick] (-1,1) rectangle (0,2); \draw[thick] (-2,2) rectangle (-1,3); \draw[thick] (2,1) rectangle (3,2); \draw[thick] (3,2) rectangle (4,3); \draw[thick] (-1,3) rectangle (0,4); \draw[thick] (2,3) rectangle (3,4); \draw[thick] (0,4) rectangle (1,5); \draw[thick] (1,4) rectangle (2,5); \end{tikzpicture}"); [/asy]

Evaluate the definite integral $\displaystyle \int^\pi_0\frac{1}{a^2-2a\cos(x)+1}dx$ for $a \in (0,1)$.

In triangle $ABC$, the perimeter is $36$ units, and the length of side $BC$ is $9$ units. Point $M$ is the midpoint of side $AC$, and $I$ is the incenter of the triangle. Determine the measure of angle $MIC$.

Determine the value of the expression [(2+0+2+4)+\left(2^0+2^4\right)+\left(2^{0^{2^4}}\right).]