Found problems: 85335
In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.
The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$
\[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]
Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$; what is the sum of all possible distances from $P$ to line $AB$?
Let $b=\log_53$. What is $\log_b\left(\log_35\right)$?
$\text{(A) }-1\qquad\text{(B) }-\frac{3}{5}\qquad\text{(C) }0\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }1$
Let $n$ be an integer greater than $1$ such that $n$ could be represented as a sum of the cubes of two rational numbers, prove that $n$ is also the sum of the cubes of two non-negative rational numbers.
Proposed by [i]Navid Safaei[/i]
Given that $a_{n}= \binom{200}{n} \cdot 6^{\frac{200-n}{3}} \cdot (\dfrac{1}{\sqrt{2}})^n$ ($ 1 \leq n \leq 95$). How many integers are there in the sequence $\{a_n\}$?
Let $n \in N^*$ . Let $a_1, a_2..., a_n$ be real such that $a_1 + a_2 +...+ a_n \ge 0$.
Prove the inequality $\sqrt{a_1^2+1}+\sqrt{a_2^2+1}+...+\sqrt{a_1^2+1}\ge \sqrt{2n(a_1 + a_2 +...+ a_n )}$.
Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?
On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$
The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img]
a) Prove that $KLMN$ is a square.
b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$.
(Alexander Shkolny)
Given $1000000000$ first natural numbers. We change each number with the sum of its digits and repeat this procedure until there will remain $1000000000$ one digit numbers. Is there more "$1$"-s or "$2$"-s?
Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$.
[i]Proposed by Mohsen Jamali[/i]
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
In a game, the first player paints a point on the plane red; the second player paints 10 uncoloured points on the plane green; then the first player paints an uncoloured point on the plane red; the second player paints 10 uncoloured points on the plane green; and so on. The first player wins if there are three red points which form an equilateral triangle. Can the second player prevent the first player from winning?
(A Kanel)
Let $n$ be a positive integer, and let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word, consisting of just letters $a$ and/or $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^n$.
A finite nonempty word $U$ is said to [i]appear[/i] in $W$ if there exist indices $k \leq \ell$ such that $U=x_k x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called [i]ubiquitous[/i] if the four words $Ua$, $Ub$, $aU$, and $bU$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words.
[i]Proposed by Grigory Chelnokov, Russia[/i]
Suppose that $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
$\frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7=$
$\textbf{(A) }-\frac{1}{64}\qquad\textbf{(B) }-\frac{1}{16}\qquad\textbf{(C) }0\qquad\textbf{(D) }\frac{1}{16}\qquad\textbf{(E) }\frac{1}{64}$
Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.
Given an acute triangle $ABC$ and points $D$, $E$, $F$ on sides $BC$, $CA$ and $AB$, respectively. If the lines $DA$, $EB$ and $FC$ are the angle bisectors of triangle $DEF$, prove that the three lines are the altitudes of triangle $ABC$.
There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.
For which of the following values of $ k$ does the equation $ \frac{x\minus{}1}{x\minus{}2}\equal{}\frac{x\minus{}k}{x\minus{}6}$ have no solution for $ x$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
In a triangle $ ABC$ with $ \angle A\equal{}90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.
Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.
[i]Proposed by Dusan Dukic, Serbia[/i]
Let $\{k_1, k_2, \dots , k_m\}$ a set of $m$ integers. Show that there exists a matrix $m \times m$ with integers entries $A$ such that each of the matrices $A + k_jI, 1 \leq j \leq m$ are invertible and their entries have integer entries (here $I$ denotes the identity matrix).
Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that
\[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\]
Find all $n$ for which equality can be attained.
[i]Proposed by Aleksandar Ilic[/i]