Found problems: 85335
Among the five statements on real numbers below, how many of them are correct?
"If $a < b < 0$ then $a < b^2$" ,
"If $0 < a < b$ then $a < b^2$",
"If $a^3 < b^3$ then $a < b$",
"If $a^2 < b^2$ then $a < b$",
"If $|a| < |b|$ then $a < b$",
(A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.
Graph Airlines $ (GA)$ operates flights between some of the cities of the Republic of Graphia. There are at least three $ GA$ flights from each city, and it is possible to travel from any city in Graphia to any city in Graphia using $ GA$ flights. $ GA$ decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel between any two cities using $ GA$ flights, yet at least $ 2/9$ of the cities have only one flight.
Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$. (For example, $A = \{1,2,3\}$, then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$.)
A subset $F$ of P $(A)$ is called [i]strong [/i] if the following is true:
If $B_1$ and $B_2$ are elements of $F$, then $B_1 \cup B_2$ is also an element of $F$.
Suppose that $F$ and $G$ are strong subsets of $P (A)$.
a) Is the union $F \cup G$ necessarily strong?
b) Is the intersection $F \cap G$ necessarily strong?
Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$
Find the number of unordered choices of $ k $ lists, each having $ m $ distinct ordered objects, among a number of $ mn $ objects.
[i]Cătălin Țigăeru[/i]
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
Let $AD$ be the altitude, $AE$ be the median, and $O$ be the circumcenter of a triangle $ABC.$ Points $X$ and $Y$ are selected inside the triangle such that $\angle BAX=\angle CAY,$ $OX\perp AX,$ and $OY\perp AY.$ Prove that points $D,E,X,Y$ are concyclic. (M. Kurskiy)
Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent.
(S. Arkhipov)
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying
\[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\]
for all integers $a$ and $b$
If $x, y$ and $2x + \frac{y}{2}$ are not zero, then
\[ \left( 2x + \frac{y}{2} \right)\left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right] \]
equals
$\textbf{(A) }1\qquad\textbf{(B) }xy^{-1}\qquad\textbf{(C) }x^{-1}y\qquad\textbf{(D) }(xy)^{-1}\qquad \textbf{(E) }\text{none of these}$
Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$ where $x$, $y$ and $z$ are integers
Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If
\[
S = \sum_{i = 1}^\infty r_i,
\]
then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$.
[i]Proposed by Aaron Lin[/i]
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.
Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than $1.$ Show that $K$ lies in a circle of radius $\frac{1}{\sqrt{3}}.$
suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that
$\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$
$\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$
$\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$
Proposed by Matin Yousefi
Curves $A,B,C$ and $D$ are defined in the plane as follows:
\begin{align*}
A &= \left\{ (x,y): x^2-y^2 = \frac{x}{x^2+y^2} \right\}, \\
B &= \left\{ (x,y): 2xy + \frac{y}{x^2+y^2} = 3 \right\}, \\
C &= \left\{ (x,y): x^3-3xy^2+3y=1 \right\}, \\
D &= \left\{ (x,y): 3x^2 y - 3x - y^3 = 0\right\}.
\end{align*}
Prove that $A \cap B = C \cap D$.
Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types:
(a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$.
(b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$.
Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves.
[i]D. Fon-der-Flaas[/i]
Show that the length of a cycle that contains every edge of a connected graph is at most the sum between the vertices and nodes of the graph, minus $ 1. $
Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.
Line segment $\overline{AE}$ of length $17$ bisects $\overline{DB}$ at a point $C$. If $\overline{AB} = 5$, $\overline{BC} = 6$ and $\angle BAC = 78^o$ degrees, calculate $\angle CDE$.