Found problems: 85335
Let $-1 \le x, y \le 1$. Prove the inequality $$2\sqrt{(1- x^2)(1 - y^2) } \le 2(1 - x)(1 - y) + 1 $$
A graph has $N$ vertices. An invisible hare sits in one of the vertices. A group of hunters tries to kill the hare. In each move all of them shoot simultaneously: each hunter shoots at a single vertex, they choose the target vertices cooperatively. If the hare was in one of the target vertices during a shoot, the hunt is finished. Otherwise the hare can stay in its vertex or jump to one of the neighboring vertices.
The hunters know an algorithm that allows them to kill the hare in at most $N!$ moves. Prove that then there exists an algorithm that allows them to kill the hare in at most $2^N$ moves.
Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation
\[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]
Suppose that the 32 computers in a certain network are numbered with the 5-bit integers $00000, 00001, 00010, ..., 11111$ (bit is short for binary digit). Suppose that there is a one-way connection from computer $A$ to computer $B$ if and only if $A$ and $B$ share four of their bits with the remaining bit being $0$ at $A$ and $1$ at $B$. (For example, $10101$ can send messages to $11101$ and to $10111$.) We say that a computer is at level $k$ in the network if it has exactly $k$ 1’s in its label $(k = 0, 1, 2, ..., 5)$. Suppose further that we know that $12$ computers, three at each of the levels $1$, $2$, $3$, and $4$, are malfunctioning, but we do not know which ones. Can we still be sure that we can send a message from $00000$ to $11111$?
Let $p_i$ denote the $i^{\text{th}}$ prime number. For any positive integer $k$ let $a_k$ denote the number of positive integers $t$ such that $p_tp_{t+1}$ divides $k.$ Let $n$ be an arbitrary positive integer. Prove that \[a_1+a_2+\cdots+a_n<\frac{n}{3}.\]
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.
[i]Proposed by Mohsen Jamali, Iran[/i]
Find all possible values of $n$ for which a rectangular board $9\times n$ can be partitioned into tiles of the shape:
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8wLzdjM2Y4ZmE0Zjg1YWZlZGEzNTQ1MmEyNTc3ZjJkNzBlMjExYmY1LnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yMiBhdCA1LjEzLjU3IEFNLnBuZw==[/img]
Construct a triangle given two of its side lengths if it is known that the median drawn from their common vertex divides the angle between them in the ratio $1:2$.
(V. Chikin)
In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students $A$, $B$, and $C$, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter.
[i]Proposed by Kevin Zhao[/i]
Let $\omega_1$, $\omega_2$, $\omega_3$ are three externally tangent circles, with $\omega_1$ and $\omega_2$ tangent at $A$. Choose points $B$ and $C$ on $\omega_1$ so that lines $AB$ and $AC$ are tangent to $\omega_3$. Suppose the line $BC$ intersect $\omega_3$ at two distinct points, and $X$ is the intersection further away to $B$ and $C$ than the other one.
Prove that one of the tangent lines of $\omega_2$ passing through $X$, is also tangent to an excircle of triangle $ABC$.
[i]Proposed by Ivan Chan Kai Chin[/i]
$100$ deputies formed $450$ commissions. Each two commissions has no more than three common deputies, and every $5$ - no more than one. Prove that, that there are $4$ commissions that has exactly one common deputy each.
$\triangle{ABC}$ is isosceles with $AB = AC >BC$. Let $D$ be a point in its interior such that $DA = DB+DC$. Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$, and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$. Prove that $B,C,P,Q$ are concyclic.
In $\triangle ABC$, $P,Q,R$ divides the perimeter of $\triangle ABC$ into three equal parts. $P,Q\in AB$. Prove that $\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}$.
We throw $ N$ balls into $ n$ urns, one by one, independently and uniformly. Let $ X_i\equal{}X_i(N,n)$ be the total number of balls in
the $ i$th urn. Consider the random variable \[ y(N,n)\equal{}\min_{1 \leq i \leq n}|X_i\minus{}\frac Nn|.\] Verify the following three statements:
(a) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow \infty$, then \[ P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}<x \right)
\rightarrow 1\minus{}e^{\minus{}x\sqrt{2/ \pi}} \;\textrm{for all}\ \; x>0 \ .\]
(b) If $ n\rightarrow \infty$ and $ N/n^3 \leq K$ ($ K$ constant), then for any $ \varepsilon > 0$ there is an $ A > 0$ such that \[ P(y(N,n) < A) > 1\minus{}\varepsilon .\]
(c) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow 0$ then \[ P(y(N,n) < 1) \rightarrow 1.\]
[i]P. Revesz[/i]
Let $O$ is a point in a plane $P$ and let $[OX,[OY,[OZ$ is distinct ray in $P$. Prove that, if $A \in [OX$ , $B \in [OY$ and $C \in [OZ$ points such that $\triangle OAB$ , $\triangle OBC$ and $\triangle OCA$ 's perimeter is 2, there is only one $(A,B,C)$ triple
Consider a pyramid whose base is a $2013$-sided polygon. On each face of the pyramid the number $0$ is written. The following operation is carried out: a vertex is chosen from the pyramid and add or subtract $1$ from all the faces that contain that vertex. It's possible, after repeating a finite number of times the previous procedure, that all the faces of the pyramid have the number $1$ written?
In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20, ...$ The number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:
$ \textbf{(A)}\ \text{two consecutive digits} \qquad\textbf{(B)}\ \text{two non-consecutive digits} \qquad$
$\textbf{(C)}\ \text{three consecutive digits} \qquad\textbf{(D)}\ \text{three non-consecutive digits} \qquad$
$\textbf{(E)}\ \text{four digits} $
Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$.
[i]Proposed by Anthony Yang and Andy Xu[/i]
Let $I$ and $O$ be respectively the incentre and circumcentre of a triangle $ABC$. If $AB = 2$, $AC = 3$ and $\angle AIO = 90^{\circ}$, find the area of $\triangle ABC$.
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)
[i]Proposed by Sergey Berlov, Russia[/i]
Let $f(n)=\frac{n}{3}$ if $n$ is divisible by $3$ and $f(n)=4n-10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c)=2$. (Here $f^5(x)$ means $f(f(f(f(f(x)))))$.)
[i]Proposed by Justin Stevens[/i]
Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.
A group of 6 math students is staying at a mathematical hotel to participate in a math tournament that will take place in the city in the coming days. This group, composed of 3 women and 3 men, was assigned rooms in a specific way by the hotel administration: in separate rooms and alternating between genders, specifically: woman, man, woman, man, woman, man, occupying the last 6 rooms in a corridor numbered from 101 to 110.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
M & H & M & H & M & H & & & & \\ \hline
110 & 109 & 108 & 107 & 106 & 105 & 104 & 103 & 102 & 101 \\ \hline
\end{tabular}
Against the hotel's rules, the group devised the following game: A valid room exchange occurs when two students in consecutive rooms move to two empty rooms, such that the difference between their new room numbers and their original ones is the same. For example, if the students in rooms 105 and 106 move to rooms 101 and 102, this would be a valid exchange since both numbers decreased by 4 units.
Determine if, following these rules, the students can manage to have rooms 101, 102, and 103 occupied by men and rooms 104, 105, and 106 occupied by women in just 3 valid exchanges.
Evaluate
\[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]
Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\dots,20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.
Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)