For what real values of $a$ the equation $\frac{2^{2x}}{2^{2x}+2^{x+1}+1}+a \frac{2^x}{2^x+1}+(a-1) = 0$ has a single root ?

If $f(x)$ is the generating function of the sequence $a_1,a_2,\ldots$ and if $f(x)=\frac{r(x)}{s(x)}$ holds such that $r(x)$ and $s(x)$ are polynomials show that $a_n$ has a homogenous recurrence.

Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.

[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1[/url] The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$ [b]8.2[/b] In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle. [url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3[/url] Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities? [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4*[/url] (asterisk problems in separate posts) [url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5[/url] Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers. [url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6[/url] Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.

Show that the equation $a^2+b^2+c^2+d^2=a^2\cdot b^2\cdot c^2\cdot d^2$ has no solution in positive integers.

The diagonals of a convex quadrilateral $ABCD$ are perpendicular to each other and intersect at the point $O$. The sum of the inradii of triangles $AOB$ and $COD$ is equal to the sum of the inradii of triangles $BOC$ and $DOA$. $(a)$ Prove that $ABCD$ has an incircle. $(b)$ Prove that $ABCD$ is symmetric about one of its diagonals.

A pile contains $2021^{2021}$ stones. In a move any pile can be divided into two piles so that the numbers of stones in them differ by a power of $2$ with non-negative integer exponent. After some move it turned out that the number of stones in each pile is a power of $2$ with non-negative integer exponent. Prove that the number of moves performed was even.

Chip and Dale play the following game. Chip starts by splitting $1001$ nuts between three piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $1001$. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly $N$ nuts in any one or more piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).

The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is $75^\circ$, then the largest angle is $\textbf{(A) }95^\circ\qquad \textbf{(B) }100^\circ\qquad \textbf{(C) }105^\circ\qquad \textbf{(D) }110^\circ\qquad \textbf{(E) }115^\circ$

Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the $2n$ points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively). [/list]

[u]Round 6 [/u] [b]p16.[/b] Triangle $ABC$ with $AB < AC$ is inscribed in a circle. Point $D$ lies on the circle and point $E$ lies on side $AC$ such that $ABDE$ is a rhombus. Given that $CD = 4$ and $CE = 3$, compute $AD^2$. [b]p17.[/b] Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of $1$ m/s. At any positive time $t$ (in seconds),Wam walks with a speed of $1$ m/s with a direction of $t$ radians clockwise of the positive $x$-axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly $5\pi$ seconds. [b]p18.[/b] Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to $8$ times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins. [u]Round 7 [/u] [b]p19.[/b] Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are $4$ rounds, of which everyone must attend exactly $2$. Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly $3$ must attend every round. In how many different ways can Ephram organize the teams like this? [b]p20.[/b] For some $4$th degree polynomial $f (x)$, the following is true: $\bullet$ $f (-1) = 1$. $\bullet$ $f (0) = 2$. $\bullet$ $f (1) = 4$. $\bullet$ $f (-2) = f (2) = f (3)$. Find $f (4)$. [b]p21.[/b] Find the minimum value of the expression $\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29}$ over all real $x$. [u]Round 8 [/u] [b]p22.[/b] Let $O$ and $I$ be the circumcenter and incenter, respectively, of $\vartriangle ABC$ with $AB = 15$, $BC = 17$, and $C A = 16$. Let $X \ne A$ be the intersection of line $AI$ and the circumcircle of $\vartriangle ABC$. Find the area of $\vartriangle IOX$. [b]p23.[/b] Find the sum of all integers $x$ such that there exist integers $y$ and $z$ such that $$x^2 + y^2 = 3(2016^z )+77.$$ [b]p24.[/b] Evaluate $$ \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor$$ [u]Round 9[/u] [b]p25.[/b]Either: 1. Submit $-2$ as your answer and you’ll be rewarded with two points OR 2. Estimate the number of teams that choose the first option. If your answer is within $1$ of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points. [b]p26.[/b] Jeff is playing a turn-based game that starts with a positive integer $n$. Each turn, if the current number is $n$, Jeff must choose one of the following: 1. The number becomes the nearest perfect square to $n$ 2. The number becomes $n-a$, where $a$ is the largest digit in $n$ Let $S(k)$ be the least number of turns Jeff needs to get from the starting number $k$ to $0$. Estimate $$\sum^{2023}_{k=1}S(k).$$ If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Estimate the smallest positive integer n such that if $N$ is the area of the $n$-sided regular polygon with circumradius $100$, $10000\pi -N < 1$ is true. If your estimation is $E$ and the actual answer is $A$, you will receive $ \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor.$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167360p28825713]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$ , $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$ . Now Suppose $\angle B$=$2\angle C$ and $CD=AB$ . Prove that $\angle A=72^0$.

Find the sum of the distinct prime factors of $512512$.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Let $ABCD$ be a cyclic quadrilateral such that $\angle BAD \le \angle ADC$. Prove that $AC + CD \le AB + BD$.

[list=a] [*]Is there a non-linear integer-coefficient polynomial $P(x)$ and an integer $N{}$ such that all integers greater than $N{}$ may be written as the greatest common divisor of $P(a){}$ and $P(b){}$ for positive integers $a>b$? [*]Is there a non-linear integer-coefficient polynomial $Q(x)$ and an integer $M{}$ such that all integers greater than $M{}$ may be written as $Q(a) - Q(b)$ for positive integers $a>b$? [/list][i]Proposed by Dylan Toh[/i]

There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?

For a real number $t$ and positive real numbers $a,b$ we have \[2a^2-3abt+b^2=2a^2+abt-b^2=0\] Find $t.$

Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.

On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets. If the expected number of times he writes the number $1$ can be written in the form $\tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i] Proposed by David Altizio [/i]

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$. [i]Proposed by Vincent Huang and Nathan Weckwerth