Found problems: 85335
Prove the inequality for all positive integer $n$ :
\[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]
Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$
Triangle $ABC$ satisfies $\angle C=90^{\circ}$. A circle passing $A,B$ meets segment $AC$ at $G(\neq A,C)$ and it meets segment $BC$ at point $D(\neq B)$. Segment $AD$ cuts segment $BG$ at $H$, and let $l$, the perpendicular bisector of segment $AD$, cuts the perpendicular bisector of segment $AB$ at point $E$. A line passing $D$ is perpendicular to $DE$ and cuts $l$ at point $F$. If the circumcircle of triangle $CFH$ cuts $AC$, $BC$ at $P(\neq C),Q(\neq C)$ respectively, then prove that $PQ$ is perpendicular to $FH$.
Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$, respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$.
In a convex quadrilateral $ABCD$, $\angle ABC = \angle CDA$. Let $X \neq C$ be the intersection of the circumcircle of $\triangle BCD$ and circle with diameter $AC$. Prove that the tangent to the circumcircle of $\triangle BCD$ at $X$, the tangent to the circumcircle of $\triangle ABD$ at $A$ concur on $BD$.
Gus has to make a list of $250$ positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if $15$ is a number from the list so the list contains $15$ numbers other than $15$. Determine the maximum number of distinct numbers the Gus list can contain.
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
Prove that if a cyclic polygon with an odd number of sides has all angles equal, then this polygon is regular.
Let $f:\mathbb{R}\to (0;+\infty )$ be a continuous function such that $\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0.$
a) Prove that $f(x)$ has the maximum value on $\mathbb{R}.$
b) Prove that there exist two sequeneces $({{x}_{n}}),({{y}_{n}})$ with ${{x}_{n}}<{{y}_{n}},\forall n=1,2,3,...$ such that they have the same limit when $n$ tends to infinity and $f({{x}_{n}})=f({{y}_{n}})$ for all $n.$
In a group of $199$ persons, each person is a friend of exactly $100$ other persons in the group. All friendships are mutual, and we do not count a person as a friend of himself/herself. For which integers $k > 1$ is the existence of $k$ persons, all being friends of each other, guaranteed?
$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.
The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
In acute triangle $ABC$, let $\odot O$ be its circumcircle, $\odot I$ be its incircle. Tangents at $B,C$ to $\odot O$ meet at $L$, $\odot I$ touches $BC$ at $D$. $AY$ is perpendicular to $BC$ at $Y$, $AO$ meets $BC$ at $X$, and $OI$ meets $\odot O$ at $P,Q$. Prove that $P,Q,X,Y$ are concyclic if and only if $A,D,L$ are collinear.
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]
In $\vartriangle PQR$, $\angle Q+10 = \angle R$. Let $M$ be the midpoint of $\overline{QR}$. If $m\angle PMQ = 100^o$, then find the measure of $\angle Q$ in degrees.
[u]Round 1[/u]
[b]p1.[/b] The fractal T-shirt for this year's Duke Math Meet is so complicated that the printer broke trying to print it. Thus, we devised a method for manually assembling each shirt - starting with the full-size 'base' shirt, we paste a smaller shirt on top of it. And then we paste an even smaller shirt on top of that one. And so on, infinitely many times. (As you can imagine, it took a while to make all the shirts.) The completed T-shirt consists of the original 'base' shirt along with all of the shirts we pasted onto it. Now suppose the base shirt requires $2011$ $cm^2$ of fabric to make, and that each pasted-on shirt requires $4/5$ as much fabric as the previous one did. How many $cm^2$ of fabric in total are required to make one complete shirt?
[b]p2.[/b] A dog is allowed to roam a yard while attached to a $60$-meter leash. The leash is anchored to a $40$-meter by $20$-meter rectangular house at the midpoint of one of the long sides of the house. What is the total area of the yard that the dog can roam?
[b]p3.[/b] $10$ birds are chirping on a telephone wire. Bird $1$ chirps once per second, bird $2$ chirps once every $2$ seconds, and so on through bird $10$, which chirps every $10$ seconds. At time $t = 0$, each bird chirps. Define $f(t)$ to be the number of birds that chirp during the $t^{th}$ second. What is the smallest $t > 0$ such that $f(t)$ and $f(t + 1)$ are both at least $4$?
[u]Round 2[/u]
[b]p4.[/b] The answer to this problem is $3$ times the answer to problem 5 minus $4$ times the answer to problem 6 plus $1$.
[b]p5.[/b] The answer to this problem is the answer to problem 4 minus $4$ times the answer to problem 6 minus $1$.
[b]p6.[/b] The answer to this problem is the answer to problem 4 minus $2$ times the answer to problem 5.
[u]Round 3[/u]
[b]p7.[/b] Vivek and Daniel are playing a game. The game ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six rounds?
[b]p8.[/b] What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$?
[b]p9.[/b] Let $U(k)$ be the set of complex numbers $z$ such that $z^k = 1$. How many distinct elements are in the union of $U(1),U(2),...,U(10)$?
[u]Round 4[/u]
[b]p10.[/b] Evaluate $29 {30 \choose 0}+28{30 \choose 1}+27{30 \choose 2}+...+0{30 \choose 29}-{30\choose 30}$. You may leave your answer in exponential format.
[b]p11.[/b] What is the number of strings consisting of $2a$s, $3b$s and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$ and $c$ is not immediately followed by $a$?
[b]p12.[/b] Compute $\left(\sqrt3 + \tan (1^o)\right)\left(\sqrt3 + \tan (2^o)\right)...\left(\sqrt3 + \tan (29^o)\right)$.
[u]Round 5[/u]
[b]p13.[/b] Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs?
[b]p14.[/b] Compute $$\sum^{2011}_{n=1}\frac{n + 4}{n(n + 1)(n + 2)(n + 3)}$$
[b]p15.[/b] Find a polynomial in two variables with integer coefficients whose range is the positive real numbers.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.
Ivan has a $m \times n$ board, and he color some squares black, so that no three black squares form a L-triomino up to rotations and reflections. What is the maximal number of black squares that Ivan can color?
[i]Proposed by Ivan Chan Kai Chin[/i]
Find six points $A_1, A_2, \ldots , A_6$ in the plane, such that for each point $A_i, i = 1, 2, \ldots , 6$ there are exactly three of the remaining five points exactly $1$ cm from $A_i$.
A triangle is said to be the Heron triangle if it has
integer sides and integer area. In a Heron triangle, the sides a; b; c satisfy
the equation b=a(a-c). Prove that the triangle is isosceles.
[b]7.[/b] Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$, that is, for every $f \in V$,
$\int_{0}^{1} f(x)P(x)dx =0$
([b]F.39[/b])
[A. Pinkus, V. Totik]
Let $f(x)=\prod_{n=1}^{\infty} \left( 1 + \frac{x}{2^n} \right)$. Show that at the point $x=1$, $f(x)$ and all its derivatives are irrational.
Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$
Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $
[i]Constantin Nicolau[/i]
Let $N_6$ be the answer to problem 6.
Given positive integers $n$ and $a$, the $n$[i]th tetration of[/i] $a$ is defined as
\[ ^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}. \]
For example, $^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536$. Compute the units digit of $^{2024}N_6$.