Found problems: 85335
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given
a) $A=\sqrt{2}$;
b) $A=1$;
c) $A=2$,
where only non-negative real numbers are admitted for square roots?
Triangle $\vartriangle ABC$ is isosceles with $AC = AB$, $BC = 1$, and $\angle BAC = 36^o$. Let $\omega$ be a circle with center B and radius $r_{\omega}= \frac{P_{ABC}}{4}$, where $P_{ABC}$ denotes the perimeter of $\vartriangle ABC$. Let $\omega$ intersect line $AB$ at $P$ and line $BC$ at $Q$. Let $I_B$ be the center of the excircle with of $\vartriangle ABC$ with respect to point $B$, and let $BI_B$ intersect $P Q$ at $S$. We draw a tangent line from $S$ to $\odot I_B$ that intersects $\odot I_B$ at point $T$. Compute the length of ST.
A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is:
$\textbf{(A)}\ y+3x-4=0 \qquad
\textbf{(B)}\ y+3x+4=0 \qquad
\textbf{(C)}\ y-3x-4=0 \qquad\\
\textbf{(D)}\ 3y+x-12=0 \qquad
\textbf{(E)}\ 3y-x-12=0 $
Constant points $B$ and $C$ lie on the circle $\omega$. The point middle of $BC$ is named $M$ by us. Assume that $A$ is a variable point on the $\omega$ and $H$ is the orthocenter of the triangle $ABC$. From the point $H$ we drop a perpendicular line to $MH$ to intersect the lines $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that with the movement of $A$ on the $\omega$, the orthocenter of the triangle $AXY$ also moves on a circle.
How many sequences of integers $1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015$ that satisfy $a_i \equiv i^2$ (mod $12$) for all $1 \le i \le 11$ are there?
(Le Anh Vinh)
A rectangle $ABCD$ and its circumcircle are given. Let $E$ be an arbitrary point on the minor arc $BC$. The tangent to the circle at $B$ meets $CE$ at point $G$. The segments $AE$ and $BD$ meet at point $K$. Prove that $GK$ and $AD$ are perpendicular.
$ \lim_{n\to\infty } \frac{1}{2^n}\left( \left( \frac{a}{a+b}+\frac{b}{b+c} \right)^n +\left( \frac{b}{b+c}+\frac{c}{c+a} \right)^n +\left( \frac{c}{c+a}+\frac{a}{a+b} \right)^n \right) ,\quad a,b,c>0 $
Let $x,y,z$ be nonnegative positive integers.
Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$
a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$.
b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.
Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.
Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.
Given that the equation $(m^2-12)x^4-8x^2-4=0$ has no real roots, then the largest value of $m$ is $p\sqrt{q}$, where $p$ and $q$ are natural numbers, $q$ is square-free. Determine $p+q$.
(A)$4$
(B)$5$
(C)$3$
(D)$6$
A natural number $k>1$ is given. The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,where $a>k + 1$. Prove that at least one of the numbers $a - 1$ or $a + 1$ composite.
Let be two nonnegative real numbers $ a,b, $ not both $ 0, $ satisfying $ \frac{1}{2a+b} +\frac{1}{a+2b} =1. $ Prove the following inequalities and explain the equality cases for [b]a),b).[/b]
[b]a)[/b] $ 4/3\le a+b\le 3/2 $
[b]b)[/b] $ 8/9\le a^2+b^2\le 9/4 $
[b]c)[/b] $ ab<1/2 $
[i]Laurențiu Panaitopol[/i]
At each vertex of a regular $14$-gon, lies a coin. Initially $7$ coins are heads, and $7$ coins are tails. Determine the minimum number $t$ such that it’s always possible to turn over at most $t$ of the coins so that in the resulting $14$-gon, no two adjacent coins are both heads and no two adjacent coins are both tails.
Find all functions \(f:\mathbb{R} \rightarrow \mathbb{R}\) such that
\[ f(x^2+y^2f(x)) = xf(y)^2-f(x)^2 \]
for every \(x, y \in \mathbb{R}\)
Let $x>0$ be a real number and $A$ a square $2\times 2$ matrix with real entries such that $\det {(A^2+xI_2 )} = 0$.
Prove that $\det{ (A^2+A+xI_2) } = x$.
In how many zeroes does the number $\dfrac{2002!}{(1001!)^2}$ end?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }200\qquad\textbf{(E) }400$
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.
Consider the set $S = \{1, 2, . . . , 2015\}$. How many ways are there to choose $2015$ distinct (possibly empty and possibly full) subsets $X_1, X_2, . . . , X_{2015}$ of $S$ such that $X_i$ is strictly contained in $X_{i+1}$ for all $1 \le i \le 2014$?
Find all functions $f: \mathbb N \to \mathbb N$ for which
\[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\]
holds for all positive integers $n$.
Define a [i]modern artwork[/i] to be a nonempty finite set of rectangles in the Cartesian coordinate plane with positive areas, pairwise disjoint interiors, and sides parallel to the coordinate axes. For a modern artwork $S$, define its [i]price[/i] to be the minimum number of colors with which Sean could paint the interiors of rectangles in $S$ such that every rectangle's interior is painted in exactly one color and every two distinct touching rectangles have distinct colors, where two rectangles are [i]touching[/i] if they share infinitely many points. For a positive integer $n$, let $g(n)$ denote the maximum price of any modern artwork with exactly $n$ rectangles. Compute $g(1) + g(2) + \cdots + g(2019).$
[i]Proposed by Yang Liu and Edward Wan[/i]
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]