Positive integers $n$ and $k$ satisfying $n\geq 2k+1$ are known to Alice. There are $n$ cards with numbers from $1$ to $n$, randomly shuffled as a deck, face down. On her turn, she does the following in order: (i) She first flips over the top card of the deck, and puts it face up on the table. (ii) Then, if Alice has not signed any card, she can sign the newest card now. The game ends after $2k+1$ turns, and Alice must have signed on some card. Let $A$ be the number on the signed cards, and $M$ be the $(k+1)^{\textup{st}}$ largest number among all $2k+1$ face-up cards. Alice's score is $|M-A|$, and she wants the score to be as close to zero as possible. For each $(n,k)$, find the smallest integer $d=d(n,k)$ such that Alice has a strategy to guarantee her score no greater than $d$. [i]Proposed by usjl[/i]

It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.

Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: [b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$. [b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$. [i]Proposed by Nikola Velov[/i]

[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle inscribed in the circle $(O)$, with orthocenter $H$. Let d be an arbitrary line which passes through $H$ and intersects $(O)$ at $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. Lines $A'P$ and $A'Q$ meet $BC$ at $K$ and $L$, respectively. Prove that $O, K, L$ and $A'$ are concyclic.

Find the disjoint sets $B$ and $C$ such that $B \cup C = \{1,2,..., 10\}$ and the product of the elements of $C$ equals the sum of elements of $B$.

Does there exist a solution in real numbers of the system of equations \[\left\{ \begin{array}{rcl} (x - y)(z - t)(z - x)(z - t)^2 = A, \\ (y - z)(t - x)(t - y)(x - z)^2 = B,\\ (x - z)(y - t)(z - t)(y - z)^2 = C,\\ \end{array} \right.\] when a) $A=2, B=8, C=6;$ b) $A=2, B=6, C=8.$?

Straight lines are drawn containing the sides of an unequal triangle $ABC$, its incircle $I$ circle and a its circumcircle, the center of which is not marked. Using only a ruler (without divisions), construct the symedian of the triangle (a straight line symmetrical to the median relative to the corresponding bisector), drawing no more than six lines.

The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.

If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

Solve in positive integers \[ x^y + y = y^x + x \]

$f$ be a function satisfying $2f(x)+3f(-x)=x^2+5x$. Find $f(7)$ [list=1] [*] $-\frac{105}{4}$ [*] $-\frac{126}{5}$ [*] $-\frac{120}{7}$ [*] $-\frac{132}{7}$ [/list]

Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform?

Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$

1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP \equal{} AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD \equal{} 90^\circ$.

If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$?

In a regular hexagon $ABCDEF$ of side length $8$ and center $K$, points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$. Find the area of pentagon $WBCUK$. [i]Proposed by Bradley Guo[/i]

a) Prove that the fraction $\frac{3n+5}{2n+3}$ is irreducible for every $n \in N$ b) Let $x,y$ be digits of decimal representation system with $x>0$, and $\frac{\overline{xy}+12}{\overline{xy}-3}\in N$, prove that $x+y=9$. Is the converse true?

Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.

Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear

Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.

$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.

An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^3-9x^2 + 10x + 5 = 0.$ Compute the side length of the triangle.