Found problems: 85335
Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$.
[i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]
Find all $(x,y,z)\in\mathbb R^3$ such that
$$x+y+z=xy+yz+zx=3$$
[i]Proposed by Ezra Guerrero, Francisco Morazan[/i]
Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$.
(a) Prove that $ QNO_1$ and $ QMO_2$ are similar.
(b) Find the locus of $ Q$ as $ X$ varies.
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased?
$ \textbf{(A)}\ \frac{26}{10} \qquad
\textbf{(B)}\ \frac{26^2}{10^2} \qquad
\textbf{(C)}\ \frac{26^2}{10} \qquad
\textbf{(D)}\ \frac{26^3}{10^3} \qquad
\textbf{(E)}\ \frac{26^3}{10^2}$
А positive integer $n{}$ is given. For every $x{}$ consider the sum \[Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.\]Find the difference $Q(10^n)-Q(10^n-1)$.
[i]Alexey Tolpygo[/i]
$7501$ points in a $96 \times 96$ square is marked. We call the $4 \times 4$ square without its central $2 \times 2$ square a [i]frame[/i]. Prove that there exist a frame with sides parallel to the $96 \times 96$ square (not necessarily from the grid lines) that contains at least $10$ marked points.
[i]Proposed by Negar Babashah, Shima Amirbeygie[/i] [b]Rated 5[/b]
Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of
$$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$
as well as all triples $(a,b,c)$ which attain the minimum.
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $
Show that $ a<b. $
Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles
Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2\minus{}x\minus{}1\equal{}0$. Let $ a_n\equal{}\frac{\alpha^n\minus{}\beta^n}{\alpha\minus{}\beta}$, $ n\equal{}1, 2, \cdots$.
(1) Prove that for any positive integer $ n$, we have $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}a_n$.
(2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n\minus{}2na^n$ for any positive integer $ n$.
How many ways are there to arrange the five letters P,U,M,A,C, such that the two vowels are not adjacent?
Seven cards numbered $1$ through $7$ lay stacked in a pile in ascending order from top to bottom ($1$ on top, $7$ on bottom). A shuffle involves picking a random card [i]of the six not currently on top[/i], and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after $10$ shuffles, $6$ is higher in the pile than $3$.
Let be a natural number $ n\ge 2 $ and an imaginary number $ z $ having the property that $ |z-1|=|z+1|\cdot\sqrt[n]{2} . $ Denote with $ A,B,C $ the points in the Euclidean plane whose representation in the complex plane are the affixes of $
z,\frac{1-\sqrt[n]{2}}{1+\sqrt[n]{2}} ,\frac{1+\sqrt[n]{2}}{1-\sqrt[n]{2}} , $ respectively. Prove that $ AB $ is perpendicular to $ AC. $
Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL \equal{} \angle XDC \equal{} 30^\circ$ and $ CX \equal{} O_1O_2$.
[i]
Author: Arnoldo Aguilar (El Salvador)[/i]
The angle bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point. Prove that the $\angle BAC> 45^o$.
The length of the common chord of two intersecting circles is $16$ feet. If the radii are $10$ feet and $17$ feet, a possible value for the distance between the centers of teh circles, expressed in feet, is:
$\text{(A)} \ 27 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ \sqrt{389} \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{undetermined}$
Polynomial $P(x)$ has integer coefficients. Prove, that if polynomials $P(x)$ and $P(P(P(x)))$ have common real root, they also have a common integer root.
Show that if a unit square is partitioned into two sets, then the diameter (least upper bound of the distances between pairs of points) of one of the sets is not less than $\sqrt{5} \slash 2.$ Show also that no larger number will do.
Let $\{U_{n,1},...,U_{n,n}\}_{n=1}^\infty$ be iid rv, uniformly distributed over [0,1] , and for $\alpha\geq 1$ consider the sets $\{[n^\alpha U_{n,1}],...,[n^\alpha U_{n,n}]\}$ , where [·] denotes the whole part. Prove that the elements of the sets $H_n\cap(\cup_{m=n+1}^\infty H_m)$ form an almost surely bounded sequence if and only if $\alpha>3$.
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
A figure is an equiangular parallelogram if and only if it is a
$ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $
$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.
There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.
Let $n$ be a positive integer. Using the integers from $1$ to $4n$ inclusive, pairs are to be formed such that the product of the numbers in each pair is a perfect square. Each number can be part of at most one pair, and the two numbers in each pair must be different. Determine, for each $n$, the maximum number of pairs that can be formed.