Found problems: 85335
In a rectangle with vertices $(0, 0), (0, 2), (n,0),(n, 2)$, ($n$ is a positive integer) find the number of longest paths starting from $(0, 0)$ and arriving at $(n, 2)$ which satis fy the following:
$\bullet$ At each movement, you can move right, up, left, down by $1$.
$\bullet$ You cannot visit a point you visited before.
$\bullet$ You cannot move outside the rectangle.
A point $ D$ is taken on the side $ AB$ of a triangle $ ABC$. Two circles passing through $ D$ and touching $ AC$ and $ BC$ at $ A$ and $ B$ respectively intersect again at point $ E$. Let $ F$ be the point symmetric to $ C$ with respect to the perpendicular bisector of $ AB$. Prove that the points $ D,E,F$ lie on a line.
Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then
$\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.
Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.
Determine the fractions of a fraction of the form $\frac{1}{ab}$ where $a,b$ are prime natural numbers such that $0 < a < b \le 200$ and $a + b > 200$
For a positive number $n$, let $g(n)$ be the product of all $1 \le k \le n$ such that gcd $(k, n) =1$, and say that $n > 1$ is reckless if $n$ is odd and $g(n) \equiv -1$ (mod $n$). Find the number of reckless numbers less than $50$.
Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$.
Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$.
[i](M. Berindeanu, RMC 2018 book)[/i]
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ \text{None of the above}
$
Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!
Circle $\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB+AC$ given that $BC=7$.
$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.
Given that a $60^{\circ}$ angle cannot be trisected with ruler and compass, prove that a $\frac{120^{\circ}}{n}$ angle cannot be trisected with ruler and compass for $n=1,2,\ldots$
An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
Given that in the diagram shown, $\angle ACB = 65^o$, $\angle BAC = 50^o$, $\angle BDC = 25^o$, $AB = 5$, and $AE = 1$, determine the value of $BE \cdot DE$.
[img]https://cdn.artofproblemsolving.com/attachments/1/2/130fcce1b383bc0dd005f61852d76e43956d4c.jpg[/img]
Let us denote with $S(n)$ the sum of all the digits of $n$.
a) Is there such an $n$ that $n+S(n)=1980$?
b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.
Let $[ABC]$ be an equilateral triangle on the side $1$. Determine the length of the smallest segment $[DE]$, where $D$ and $E$ are on the sides of the triangle, which divides $[ABC]$ into two figures with equal area.
Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.
[asy]
/*Using regular asymptote, this diagram would take 30 min to make. Using cse5, this takes 5 minutes. Conclusion? CSE5 IS THE BEST PACKAGE EVER CREATED!!!!*/
size(100);
import cse5;
pathpen=black;
anglefontpen=black;
pointpen=black;
anglepen=black;
dotfactor=3;
pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE;
EE=(B+C)/2;
D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle);
D(MP("$E$",EE,N)--MP("$D$",D,S));
D(D);D(EE);
MA("80^\circ",8,D,EE,C,0.1);
MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C);
draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow());
MA("100^\circ",8,A,B,C,0.1,0);
MA("60^\circ",8,C,A,B,0.1,0);
//Credit to TheMaskedMagician for the diagram
[/asy]
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals
$\textbf{(A) }\frac{1}{4}\cos 10^\circ\qquad\textbf{(B) }\frac{\sqrt{3}}{8}\qquad\textbf{(C) }\frac{1}{4}\cos 40^\circ\qquad\textbf{(D) }\frac{1}{4}\cos 50^\circ\qquad\textbf{(E) }\frac{1}{8}$
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?
Find all prime numbers $ a,b,c$ that fulfill the equality:
$ (a\minus{}2)!\plus{}2b!\equal{}22c\minus{}1$
Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.
The set $M=\{1,2,\ldots,2n\}~(n\ge2)$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $k^3+1\le n$. Prove that there exist $k+1$ even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_j$ $(1\le j\le k)$ such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_r$ $(1\le r\le k)$.
There is a $\triangle ABC$ which $\angle C=90$, and $\bar{AB}=36$
On the circumcircle of $\triangle ABC$, there is $\overarc{BC}$ which does not include point $A$.
D is on $\overarc{BC}$. It satisfies $2\times\angle CAD = \angle BAD $
$E: \bar{AD}\cap\bar{BC} $ $ \bar{AE}=20 $
Find $ \bar{BD}^2 $