Found problems: 85335
Solve the equation $ \frac{\sqrt{2+x} +\sqrt{2-x}}{\sqrt{2+x} -\sqrt{2-x}} =\sqrt 3. $
Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove:
\[ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} \]
And determine when equality holds.
Let $A$ be the set of all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(xy)=xf(y)$ for all $x,y \in \mathbb{R}$.
(a) If $f \in A$ then show that $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$
(b) For $g,h \in A$, define a function $g\circ h$ by $(g \circ h)(x)=g(h(x))$ for $x \in \mathbb{R}$. Prove that $g \circ h$ is in $A$ and is equal to $h \circ g$.
A $6\text{-year stock}$ that goes up $30\%$ in the first year, down $30\%$ in the second, up $30\%$ in the third, down $30\%$ in the fourth, up $30\%$ in the fifth, and down $30\%$ in the sixth is equivalent to a $3\text{-year stock}$ that loses $x\%$ in each of its three years. Compute $x$.
Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$, let $P$ be a point on $BC$, and let $Q$ be the point (other than $A$) where the line through $A$ and $P$ intersects the circumcircle of $ABC$. Prove that
\[PQ\le \frac{25}{4\sqrt{6}}.\]
There exists a function $G(x)$ such that $G(x)+G\left(\frac{x-\sqrt{3}}{x\sqrt{3}+1}\right)=\sqrt{2}+x$. Find $G(x)$.
[i]Proposed by beanielove2.[/i]
Alice and Bob are playing the following game: They have an $8\times8$ chessboard. Initially, all grids are white. Each round, Alice chooses a white grid and paints it black. Then Bob chooses one of the neighbors of that grid and paints it black. Or he does nothing. After that, Alice may decide to continue the game or not. The goal of Alice is to maximize the number of connected components of black grids, on the other hand, Bob wants to minimize that number. If both of them are extremely smart, how many connected components will be in the end of the game?
Prove that for any three real numbers $x,y,z$ the following inequality holds:
$$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$
What is the remainder when $7^{8^9}$ is divided by $1000?$
Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.
Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.
Celia chooses a number $n$ and writes the list of natural numbers from $1$ to $n$: $1, 2, 3, 4, ..., n-1, n.$ At each step, it changes the list: it copies the first number to the end and deletes the first two. After $n-1$ steps a single number will be written.
For example, for $n=6$ the five steps are: $$ 1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5$$
and the number $5$ is written.
Celia chose a number $n$ between $1000$ and $3000$ and after $n-1$ steps the number $1$ was written.
Determine all the values of $n$ that Celia could have chosen.
Justify why those values work, and the others do not.
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color?
$\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$
Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, what is the minimal number of connected components of the graph $G/(X \cup Y)$?
Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$. The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 8)[/i]
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where $a_{n+i}=a_i,i=1,2,\ldots,k$
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
Let $ABC$ be an acute and scalene of circumcircle $\Gamma$ and orthocenter $H$. Let $A_1,B_1,C_1$ be the second intersection points of the lines $AH, BH, CH$ with $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the intersection point of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the intersection point of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle AMP$ .
a,b,c are positive & abc=1, then show that
a^(b+c) .b^(c+a) .c^(a+b) ≤ 1
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$.
[i]Proposed by Liam Baker, South Africa[/i]
James writes down fifteen 1's in a row and randomly writes $+$ or $-$ between each pair of consecutive 1's. One such example is \[1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.\] What is the probability that the value of the expression James wrote down is $7$?
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }\dfrac{6435}{2^{14}}&\textbf{(C) }\dfrac{6435}{2^{13}}\\\\
\textbf{(D) }\dfrac{429}{2^{12}}&\textbf{(E) }\dfrac{429}{2^{11}}&\textbf{(F) }\dfrac{429}{2^{10}}\\\\
\textbf{(G) }\dfrac1{15}&\textbf{(H) } \dfrac1{31}&\textbf{(I) }\dfrac1{30}\\\\
\textbf{(J) }\dfrac1{29}&\textbf{(K) }\dfrac{1001}{2^{15}}&\textbf{(L) }\dfrac{1001}{2^{14}}\\\\
\textbf{(M) }\dfrac{1001}{2^{13}}&\textbf{(N) }\dfrac1{2^7}&\textbf{(O) }\dfrac1{2^{14}}\\\\
\textbf{(P) }\dfrac1{2^{15}}&\textbf{(Q) }\dfrac{2007}{2^{14}}&\textbf{(R) }\dfrac{2007}{2^{15}}\\\\
\textbf{(S) }\dfrac{2007}{2^{2007}}&\textbf{(T) }\dfrac1{2007}&\textbf{(U) }\dfrac{-2007}{2^{14}}\end{array}$
Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.