Found problems: 85335
There are dwarves in a forest and each one of them owns exactly 3 hats which are numbered with numbers $1, 2, \dots 28$. Three hats of a dwarf are numbered with different numbers and there are 3 festivals in this forest in a day. In the first festival, each dwarf wears the hat which has the smallest value, in the second festival, each dwarf wears the hat which has the second smallest value and in the final festival each dwarf wears the hat which has the biggest value. After that, it is realized that there is no dwarf pair such that both of two dwarves wear the same value in at least two festivals. Find the maximum value of number of dwarves.
Let $a_1,a_2,\cdots,a_n$ be a sequence of positive numbers, so that $a_1 + a_2 +\cdots + a_n \leq \frac 12$. Prove that
\[(1 + a_1)(1 + a_2) \cdots (1 + a_n) < 2.\]
[hide="Remark"]Remark. I think this problem was posted before, but I can't find the link now.[/hide]
You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place.
[i]Linus Hamilton[/i]
Suppose that $n\ge2$ and $a_1,a_2,...,a_n$ are natural numbers that $ (a_1,a_2,...,a_n)=1$. Find all strictly increasing function $f: \mathbb{Z} \to \mathbb{R} $ that:
$$ \forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})$$
[i]Proposed by Navid Safaei and Ali Mirzaei [/i]
A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$?
[i]David Yang.[/i]
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?
$\textbf{(A)}\ 9\pi \qquad\textbf{(B)}\ 12\pi \qquad\textbf{(C)}\ 18\pi \qquad\textbf{(D)}\
24\pi \qquad\textbf{(E)}\ 27\pi$
The numbers from $1$ to $50$ are printed on cards. The cards are shuffled and then laid out face up in $5$ rows of $10$ cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?
The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.
In a trapezoid $ABCD$, the side $DA$ is perpendicular to the bases $AB$ and $CD$. Also $AB=45$, $CD =20$, $BC =65$. Let $P$ be a point on the side $BC$ such that $BP=45$ and let $M$ be the midpoint of $DA$. Calculate the length of $PM$ .
Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.
Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$
If no element of $M$ is equal to the sum of any two elements of $M,$
find the least value that the greatest element of $M$ can take.
Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$.
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.
In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.
Each unit square of a $5 \times 5$ board is colored blue or yellow. Prove that there is a rectangle with sides parallel to the sides. axes of the board, such that its four corners are the same color.
Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$.
In a quadrilateral $ABCD$, a circle is inscribed that is tangent to the sides $AB, BC, CD$ and $DA$ at points $M, N, P$ and $Q$, respectively. If $(AM) (CP) = (BN) (DQ)$, prove that $ABCD$ is an cyclic quadrilateral.
In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$
For positive real numbers $a$ and $b,$ let $f(a,b)$ denote the real number $x$ such that area of the (non-degenerate) triangle with side lengths $a,b,$ and $x$ is maximized. Find
\[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]
When the expression $$(xy-5x+3y-15)^n$$ for some positive integer $n$ is expanded and like terms are combined, the expansion contains at least 2021 distinct terms. Compute the minimum possible value of $n$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 4)[/i]
Let $a_1, a_2,...,a_{51}$ be non-zero elements of a field of characteristic $p$. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_1, ... , b_{51}$. If this new sequence is a permutation of the original one, find all possible values of $p$.
Let $\gamma$ be circle and let $P$ be a point outside $\gamma$. Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$. Let $S$ be a point on $\gamma$ such that $BS \parallel QR$. Prove that $SA$ bisects $QR$.
Show that exists a sequence of $ 100$ terms such that:
1)Every term is a perfect square
2) every term is greater than the one before it ( it is strictly increasing)
3)Every two terms of the sequence are relative prime
4) The average between two consecutive terms is also a perfect square
Daniel
An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
fill(unitsquare,gray);
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);[/asy]$ \textbf{(A)}\ 80\minus{}20\pi \qquad
\textbf{(B)}\ 60\minus{}10\pi \qquad
\textbf{(C)}\ 80\minus{}10\pi \qquad
\textbf{(D)}\ 60\plus{}10\pi \qquad
\textbf{(E)}\ 80\plus{}10\pi$