Found problems: 85335
Function $f(x)=ax^2+8x+3(a<0)$. For any given nerative number $a$, define the largest positive number $l(a)$: $|f(x)|\leq5$ for all $x\in[0,l(a)]$.
Find the largest $l(a)$, and $a$ when $l(a)$ takes its maximum value.
Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.
Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that
$|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$,
with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.
Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$.
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$
(3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.
Find all real numbers for which $3^x+4^x=5^x$.
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Let $\ell$ be the tangent to $\omega$ parallel to $BC$ and distinct from $BC$. Let $D$ be the intersection of $\ell$ and $AC$, and let $M$ be the midpoint of $\overline{ID}$. Prove that $\angle AMD = \angle DBC$.
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$
Let $ABC$ be a triangle with $AB=13,BC=15,AC=14$, circumcenter $O$, and orthocenter $H$, and let $M,N$ be the midpoints of minor and major arcs $BC$ on the circumcircle of $ABC$. Suppose $P\in AB, Q\in AC$ satisfy that $P,O,Q$ are collinear and $PQ||AN$, and point $I$ satisfies $IP\perp AB,IQ\perp AC$. Let $H'$ be the reflection of $H$ over line $PQ$, and suppose $H'I$ meets $PQ$ at a point $T$. If $\frac{MT}{NT}$ can be written in the form $\frac{\sqrt{m}}{n}$ for positive integers $m,n$ where $m$ is not divisible by the square of any prime, then find $100m+n$.
[i]Proposed by Vincent Huang[/i]
Find the number of ways in which three numbers can be selected from the set $\{1,2,\cdots ,4n\}$, such that the sum of the three selected numbers is divisible by $4$.
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.
Consider a nonzero integer number $n$ and the function $f:\mathbb{N}\to \mathbb{N}$ by
\[ f(x) = \begin{cases}
\frac{x}{2} & \text{if } x \text{ is even} \\
\frac{x-1}{2} + 2^{n-1} & \text{if } x \text{ is odd}
\end{cases}.
\] Determine the set: \[
A = \{ x\in \mathbb{N} \mid \underbrace{\left( f\circ f\circ ....\circ f \right)}_{n\ f\text{'s}}\left( x \right)=x \}.
\]
Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.
If $y = 2x$ and $z = 2y$, then $x + y + z$ equals
\[ \text{(A)}\ x \qquad \text{(B)}\ 3x \qquad \text{(C)}\ 5x \qquad \text{(D)}\ 7x \qquad \text{(E)}\ 9x \]
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is $10\%$. She leaves a $15\%$ tip on the prices of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50$ for dinner. What is the cost of here dinner without tax or tip?
$ \textbf{(A)}\ \$18\qquad\textbf{(B)}\ \$20\qquad\textbf{(C)}\ \$21\qquad\textbf{(D)}\ \$22\qquad\textbf{(E)}\ \$24$
Solve in real numbers the system:
$$\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}$$
How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$?
A number plus $4$ is $2003$. What is the number?
Given a square $ABCD$ with side $1$, and a square inside $ABCD$ with side $x$, find (in terms of $x$) the radio $r$ of the circle tangent to two sides of $ABCD$ and touches the square with side $x$. (See picture).
In isosceles $\triangle ABC$, $AB=AC$, points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$, $DF\parallel AC$. The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$. Let $DE$ meet $\omega_2$ at $K\neq E$. Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$. Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.
Points $P,Q,R,S$ lie on a circle and $\angle PSR$ is right. $H,K$ are the projections of $Q$ on lines $PR,PS$. Prove that $HK$ bisects segment $ QS$.
$x,y,z,t \in \mathbb R_+^*$:
\[ (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} \]
Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2.
When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.
Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area?
(Problem from Latvia)