Let $ABC$ be a triangle, and let $P$ be the midpoint of $AC$. A circle intersects $AP, CP, BC, AB$ sequentially at their inner points $K, L, M, N$. Let $S$ be the midpoint of $KL$. Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$, then the intersection of $KN$ and $ML$ lies on the perpendicular bisector of the $PS$. (Jan Mazák)

In a game of Shipbattle, Willis secretly places his aircraft carrier somewhere in a $9 \times 9$ grid, represented by five consecutive squares. Two example positions are shown below. [asy] size(5cm); fill((2,7)--(7,7)--(7,8)--(2,8)--cycle); fill((5,1)--(6,1)--(6,6)--(5,6)--cycle); for (int i = 0; i <= 9; ++i) { draw((i,0)--(i,9)); draw((0,i)--(9,i)); } [/asy] Phyllis then takes shots at the grid, one square at a time, trying to hit Willis's aircraft carrier. What is the minimum number of shots that Phyllis must take to ensure that she hits the aircraft carrier at least once?

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a table there are $2013$ cards that have written, each one, a different integer number, from $1$ to $2013$; all the cards face down (you can't see what number they are). It is allowed to select any set of cards and ask if the average of the numbers written on those cards is integer. The answer will be true. a) Find all the numbers that can be determined with certainty by several of these questions. b) We want to divide the cards into groups such that the content of each group is known even though the individual value of each card in the group is not known. (For example, finding a group of $3$ cards that contains $1, 2$, and $3$, without knowing what number each card has.) What is the maximum number of groups that can be obtained?

Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$.

Find the smallest and largest values of the expression $$\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1}$$ (The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)

Compute the sum of all positive integers $n < 200$ such that $gcd(n, k) \ne 1$ for every $k \in\{2 \cdot 11 \cdot 19, 3 \cdot 13 \cdot 17, 5 \cdot 11 \cdot 13, 7 \cdot 17 \cdot 19\}$.

Let $a_1 < a_2 < a_3 < a_4$ be positive integers such that the following conditions hold: -$\gcd(a_i,a_j)>1$ holds for all integers $1\le i < j\le 4$. -$\gcd(a_i,a_j,a_k)=1$ holds for all integers $1\le i < j < k\le 4$. Find the smallest possible value of $a_4$. [i]Proposed by James Lin[/i]

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n$ . Find the maximum possible value of positive integer $n$ .

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

Consider all cubic polynomials $f(x)$ such that $f(2018)=2018$, the graph of $f$ intersects the $y$-axis at height $2018$, the coefficients of $f$ sum to $2018$, and $f(2019)>(2018)$. We define the infinimum of a set $S$ as follows. Let $L$ be the set of lower bounds of $S$. That is, $\ell\in L$ if and only if for all $s\in S$, $\ell\leq s$. Then the infinimum of $S$ is $\max(L)$. Of all such $f(x)$, what is the infinimum of the leading coefficient (the coefficient of the $x^3$ term)?

Let $m$ and $n{}$ be positive integers of the same parity such that $n^2-1$ divides $|m^2+1-n^2|$. Is the number $|m^2+1-n^2|$ is a perfect square?

If the length of a rectangle is increased by $20\% $ and its width is increased by $50\% $, then the area is increased by $\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% $

Find the largest possible sum $ m + n$ for positive integers $m, n \le 100$ such that $m + 1 \equiv 3$ (mod $4$) and there exists a prime number $p$ and nonnegative integer $a$ such $\frac{m^{2n-1}-1}{m-1} = m^n+p^a$ .

Suppose $x$ and $y$ are positive real numbers such that $x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$. What is the greatest possible value of $\log_2{y}$? $\textbf{(A)}3~\textbf{(B)}4~\textbf{(C)}3+\sqrt{2}~\textbf{(D)}4+\sqrt{3}~\textbf{(E)}7$

In the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$. [i]Proposed by Mahdi Etesamifard[/i]

Prove that the polynomial$$P(x)=(x-1)(x-2)(x-3)-1$$is irreducible in $\mathbb{Z}[x].$

$f$ is a polynomial of degree $n$ with integer coefficients and $f(x)=x^2+1$ for $x=1,2,\cdot ,n$. What are the possible values for $f(0)$?

Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+\dfrac{1}{a_3a_4}\cdots=\dfrac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$. [i]Individual #7[/i]

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$ is the square of an integer. [i]Proposed by Fedir Yudin[/i]

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$: $$f(mf(n)) + f(n) | mn + f(f(n)).$$

Secco and Ramon are drunk in the real line over the integer points $a$ and $b$, respectively. Our real line is a little bit special, though: the interval $(-\infty, 0)$ is covered by a sea of lava. Being aware of this fact, and also because they are drunk, they decided to play the following game: initially they choose an integer number $k>1$ using a fair dice as large as desired, and therefore they start the game. In the first round, each player writes the point $h$ for which it wants to go. After that, they throw a coin: if the result is heads, they go to the desired points; otherwise, they go to the points $2g - h$, where $g$ is the point where each of the players were in the precedent round (that is, in the first round $g = a$ for Secco and $g = b$ for Ramon). They repeat this procedure in the other rounds, and the game finishes when some of the player is over a point exactly $k$ times bigger than the other (if both of the player end up in the point $0$, the game finishes as well). Determine, in values of $k$, the initial values $a$ and $b$ such that Secco and Ramon has a winning strategy to finish the game alive. [i]Observation: If any of the players fall in the lave, he dies and both of them lose the game[/i]

$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$ [i](S. Berlov)[/i]