For each positive integer $ n$,define $ f(n)\equal{}lcm(1,2,...,n)$. (a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant. (b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for which this maximum is attained.

Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is an integer. Prove that $a+b+c+d$ is [b]not[/b] a prime number.

Consider the numbers from $1$ to $32$. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the $32$ numbers as: $(1, 2), (3,4),(5, 6), (7, 8),..., (27, 28),(29, 30), (31,32)$, we get the following list of $16$ numbers: $3,7,11,5,...,11,59,7$. where there are repetitions. The game continues in a similar way until in the end only one number remains. Determine the highest possible value from the number that remains at the end.

Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord. Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.

Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.

For each integer $n \ge 100$ we define $T(n)$ to be the number obtained from $n$ by moving the two leading digits to the end. For example, $T(12345) = 34512$ and $T(100) = 10$. Find all integers $n \ge 100$ for which $n + T(n) = 10n$.

A set of $ 25$ square blocks is arranged into a $ 5\times 5$ square. How many different combinations of $ 3$ blocks can be selected from that set so that no two are in the same row or column? $ \textbf{(A)}\ 100\qquad \textbf{(B)}\ 125\qquad \textbf{(C)}\ 600\qquad \textbf{(D)}\ 2300\qquad \textbf{(E)}\ 3600$

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

A block of mass $\text{m = 3.0 kg}$ slides down one ramp, and then up a second ramp. The coefficient of kinetic friction between the block and each ramp is $\mu_\text{k} = \text{0.40}$. The block begins at a height $\text{h}_\text{1} = \text{1.0 m}$ above the horizontal. Both ramps are at a $\text{30}^{\circ}$ incline above the horizontal. To what height above the horizontal does the block rise on the second ramp? (A) $\text{0.18 m}$ (B) $\text{0.52 m}$ (C) $\text{0.59 m}$ (D) $\text{0.69 m}$ (E) $\text{0.71 m}$

Let \[ \frac{p}{q} = \frac{2017}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\cdots}}}}}}}}}}}\] where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

$AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ so that $AP = 4$ and $BP = 3$. Points $C$ and $D$ are drawn on circle $O$ so that $OC$ bisects $AP$ and $OD$ bisects $BP$. What is the degree measure of $\angle COD$?

Let $a,b,c$ be positive reals. Prove that \[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \] [i]Calvin Deng.[/i]

Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

Let $ABCD$ be a quadrilateral such that $\angle A=\angle C=90^{\circ}$. If $A, D$ and the midpoints of $BA, BC$ are concyclic, show that the midpoints of $AD, DC$ and $B, C$ are concyclic.

A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at $1$ meter per second. Suddenly someone turns on the turntable; it spins at $30$ rpm. Consider the set $S$ of points the mouse can reach in his car within $1$ second after the turntable is set in motion. What is the area of $S$, in square meters?

Let $n$ be a positive integer. A grid of dimensions $n \times n$ is divided in $n^2$ $1 \times 1$ squares. Every segment of length $1$ (side of a square) from this grid is coloured in blue or red. The number of red segments is not greater than $n^2$. Find all positive integers $n$, for which the grid always will cointain at least one $1 \times 1$ square which has at least three blue sides.

For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that $$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$

This problem is given by my teacher. :wink: [size=120]Seven skiers numbered 1,2,3,4,5,6,7 set out in turn at the starting point,each one slides the same distance at a constant speed. During this period,everyone just had two "beyond" experience.(going beyond one skier or be went beyond by another skier is called a "beyond" experience). When the race ended,we would decide the rank according to the order that skiers reached the ending. Prove that:there are two different rank at most.[/size]

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) $A=2, B=6, C=8.$?