Evaluate $ \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx$.

[b]p1.[/b] David is taking a $50$-question test, and he needs to answer at least $70\%$ of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test? [b]p2.[/b] You decide to flip a coin some number of times, and record each of the results. You stop flipping the coin once you have recorded either $20$ heads, or $16$ tails. What is the maximum number of times that you could have flipped the coin? [b]p3.[/b] The width of a rectangle is half of its length. Its area is $98$ square meters. What is the length of the rectangle, in meters? [b]p4.[/b] Carol is twice as old as her younger brother, and Carol's mother is $4$ times as old as Carol is. The total age of all three of them is $55$. How old is Carol's mother? [b]p5.[/b] What is the sum of all two-digit multiples of $9$? [b]p6.[/b] The number $2016$ is divisible by its last two digits, meaning that $2016$ is divisible by $16$. What is the smallest integer larger than $2016$ that is also divisible by its last two digits? [b]p7.[/b] Let $Q$ and $R$ both be squares whose perimeters add to $80$. The area of $Q$ to the area of $R$ is in a ratio of $16 : 1$. Find the side length of $Q$. [b]p8.[/b] How many $8$-digit positive integers have the property that the digits are strictly increasing from left to right? For instance, $12356789$ is an example of such a number, while $12337889$ is not. [b]p9.[/b] During a game, Steve Korry attempts $20$ free throws, making 16 of them. How many more free throws does he have to attempt to finish the game with $84\%$ accuracy, assuming he makes them all? [b]p10.[/b] How many dierent ways are there to arrange the letters $MILKTEA$ such that $TEA$ is a contiguous substring? For reference, the term "contiguous substring" means that the letters $TEA$ appear in that order, all next to one another. For example, $MITEALK$ would be such a string, while $TMIELKA$ would not be. [b]p11.[/b] Suppose you roll two fair $20$-sided dice. What is the probability that their sum is divisible by $10$? [b]p12.[/b] Suppose that two of the three sides of an acute triangle have lengths $20$ and $16$, respectively. How many possible integer values are there for the length of the third side? [b]p13.[/b] Suppose that between Beijing and Shanghai, an airplane travels $500$ miles per hour, while a train travels at $300$ miles per hour. You must leave for the airport $2$ hours before your flight, and must leave for the train station $30$ minutes before your train. Suppose that the two methods of transportation will take the same amount of time in total. What is the distance, in miles, between the two cities? [b]p14.[/b] How many nondegenerate triangles (triangles where the three vertices are not collinear) with integer side lengths have a perimeter of $16$? Two triangles are considered distinct if they are not congruent. [b]p15.[/b] John can drive $100$ miles per hour on a paved road and $30$ miles per hour on a gravel road. If it takes John $100$ minutes to drive a road that is $100$ miles long, what fraction of the time does John spend on the paved road? [b]p16.[/b] Alice rolls one pair of $6$-sided dice, and Bob rolls another pair of $6$-sided dice. What is the probability that at least one of Alice's dice shows the same number as at least one of Bob's dice? [b]p17.[/b] When $20^{16}$ is divided by $16^{20}$ and expressed in decimal form, what is the number of digits to the right of the decimal point? Trailing zeroes should not be included. [b]p18.[/b] Suppose you have a $20 \times 16$ bar of chocolate squares. You want to break the bar into smaller chunks, so that after some sequence of breaks, no piece has an area of more than $5$. What is the minimum possible number of times that you must break the bar? For an example of how breaking the chocolate works, suppose we have a $2\times 2$ bar and wish to break it entirely into $1\times 1$ bars. We can break it once to get two $2\times 1$ bars. Then, we would have to break each of these individual bars in half in order to get all the bars to be size $1\times 1$, and we end up using $3$ breaks in total. [b]p19.[/b] A class of $10$ students decides to form two distinguishable committees, each with $3$ students. In how many ways can they do this, if the two committees can have no more than one student in common? [b]p20.[/b] You have been told that you are allowed to draw a convex polygon in the Cartesian plane, with the requirements that each of the vertices has integer coordinates whose values range from $0$ to $10$ inclusive, and that no pair of vertices can share the same $x$ or $y$ coordinate value (so for example, you could not use both $(1, 2)$ and $(1, 4)$ in your polygon, but $(1, 2)$ and $(2, 1)$ is fine). What is the largest possible area that your polygon can have? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two points $E$ and $F$ are randomly chosen in the interior of unit square $ABCD$. Let the line through $E$ parallel to $AB$ hit $AD$ at $E_1$, the line through $E$ parallel to $AD$ hit $CD$ at $E_2$, the line through $F$ parallel to $AB$ hit $BC$ at $F_1$, and the line through $F$ parallel to $BC$ hit $AB$ at $F_2$. The expected value of the overlap of the areas of rectangles $EE_1DE_2$ and $FF_1BF_2$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [i]Proposed by Kevin Zhao[/i]

Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.) What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)

Let $S(n)=1\varphi(1)+2\varphi(2) \ldots +n\varphi(n)$, where $\varphi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$. (For instance $\varphi(12)=4$ and $\varphi(20)=8$.) Prove that for all $n \geq 2018$, the following inequality holds: $$0.17n^3 \leq S(n) \leq 0.23n^3$$

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

$ I_a$ is the excenter of the triangle $ ABC$ with respect to $ A$, and $ AI_a$ intersects the circumcircle of $ ABC$ at $ T$. Let $ X$ be a point on $ TI_a$ such that $ XI_a^2\equal{}XA.XT$. Draw a perpendicular line from $ X$ to $ BC$ so that it intersects $ BC$ in $ A'$. Define $ B'$ and $ C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

Given is a sequence $a_1, a_2, \ldots$, such that $a_1=1$ and $a_{n+1}=\frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$. Which terms of this sequence are positive integers?

The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has: $ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$ $\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $

Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations: $$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$

Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order). $r$ is the tangent to $\Gamma$ at point A. $s$ is the tangent to $\Gamma$ at point D. Let $E=r \cap BC,F=s \cap BC$. Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$ Show that the points $X,Y,Z$ are collinear. Note: assume the existence of all above points.

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

Find the units digit of the sum \[(1!)^2+(2!)^2+(3!)^2+(4!)^2+\cdots+(2007!)^2.\] $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }5\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }9$

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]

Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles? by Morteza Saghafian

Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$. Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$. Thanks for show me the mistake typing

Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.

A group of 6 math students is staying at a mathematical hotel to participate in a math tournament that will take place in the city in the coming days. This group, composed of 3 women and 3 men, was assigned rooms in a specific way by the hotel administration: in separate rooms and alternating between genders, specifically: woman, man, woman, man, woman, man, occupying the last 6 rooms in a corridor numbered from 101 to 110. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline M & H & M & H & M & H & & & & \\ \hline 110 & 109 & 108 & 107 & 106 & 105 & 104 & 103 & 102 & 101 \\ \hline \end{tabular} Against the hotel's rules, the group devised the following game: A valid room exchange occurs when two students in consecutive rooms move to two empty rooms, such that the difference between their new room numbers and their original ones is the same. For example, if the students in rooms 105 and 106 move to rooms 101 and 102, this would be a valid exchange since both numbers decreased by 4 units. Determine if, following these rules, the students can manage to have rooms 101, 102, and 103 occupied by men and rooms 104, 105, and 106 occupied by women in just 3 valid exchanges.

Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.

Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$. [i]Proposed by Harry Kim[/i]

An $8$ by $8$ board (with $64$ $1$ by $1$ squares) is painted white. We are allowed to choose any rectangle consisting of $3$ of the $64$ squares and paint each of the $3$ squares in the opposite colour (the white ones black, the black ones white). Is it possible to paint the entire board black by means of such operations? (IS Rubanov, Kirov)

Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.