Found problems: 85335
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.
The number $27,\,000,\,001$ has exactly four prime factors. Find their sum.
Let $ABC$ be triangle with the symmedian point $L$ and circumradius $R$. Construct parallelograms $ ADLE$, $BHLK$, $CILJ$ such that $D,H \in AB$, $K, I \in BC$, $J,E \in CA$ Suppose that $DE$, $HK$, $IJ$ pairwise intersect at $X, Y,Z$. Prove that inradius of $XYZ$ is $\frac{R}{2}$ .
You have a set of five weights, together with a balance that allows you to compare the weight of two things. The weights are known to be $10$, $20$,$30$,$40$ and $50$ grams, but are otherwise identical except for their labels. The $10$ and $50$ gram weights are clearly labelled, but the labels have been erased on the remaining weights. Using the balance exactly once, is it possible to determine what one of the three unlabelled weights is? If so, explain how, and if not, explain why not.
Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and
\[\binom{3k}{k} \equiv 1 \pmod{p}.\]
Quadrilateral $ABCD$ is inscribed in a circle $O$.
Let $AB\cap CD=E$ and $P\in BC, EP\perp BC$, $R\in AD, ER\perp AD$, $EP\cap AD=Q, ER\cap BC=S$
Let $K$ be the midpoint of $QS$
Prove that $E, K, O$ are collinear.
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$
Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -
Let $a_1>a_2>.....>a_r$ be positive real numbers .
Compute $\lim_{n\to \infty} (a_1^n+a_2^n+.....+a_r^n)^{\frac{1}{n}}$
Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB\equal{}\alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ BC$, $ CA$, $ AB$, respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.
It is given rectangle $ABCD$ with length $|AB|=15cm$ and with length of altitude $|BE|=12cm$ where $BC$ is altitude of triangle $ABC$ . Find perimeter and area of rectangle $ABCD$ .
Let $P$ be a point in the interior of quadrilateral $ABCD$ such that:
$$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$
Prove that:
$$\angle PBD= \left | \angle BCA - \angle PCA \right |$$
[i]Proposed by Ali Zamani[/i]
In triangle $ ABC$, one has marked the incenter, the foot of altitude from vertex $ C$ and the center of the excircle tangent to side $ AB$. After this, the triangle was erased. Restore it.
Let $ABCD$ be a parallelogram with $\angle ABC=120^\circ$, $AB=16$ and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to
$\textbf{(A) }1\qquad
\textbf{(B) }2\qquad
\textbf{(C) }3\qquad
\textbf{(D) }4\qquad
\textbf{(E) }5$
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(16,0),C=(26,10*sqrt(3)),D=(10,10*sqrt(3)),E=(0,10*sqrt(3));
draw(A--B--C--E--B--A--D);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,N);
label("$E$",E,N);
label("$F$",extension(A,D,B,E),W);
label("$4$",(D+E)/2,N);
label("$16$",(8,0),S);
label("$10$",(B+C)/2,SE);
[/asy]
Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold:
[list=a][*]$f(1)=0$,
[*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list]
Find the value of $a^{2013}+b^{2013}+c^{2013}$.
Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$
Georg has a board displaying the numbers from $1$ to $50$. Georg may strike out a number if it can be formed by starting with the number $2$ and doing one or more calculations where he either multiplies by $10$ or subtracts $3$.
Which of the board’s numbers may Georg strike out?[img]https://cdn.artofproblemsolving.com/attachments/c/e/1bea13b691d3591d782e698bedee3235f8512f.png[/img]
Example: Georg may strike out $26$ because it may, for example, be formed by starting with $2$, multiplying by $10$, subtracting $3$ three times, multiplying by $10$ and subtracting $3$ twenty-eight times.
If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
Prove that there exist 1000 consecutive numbers such that none of them is divisible by its sum of the digits
Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_1, x_2, \ldots, x_n$ for which
[list]
[*]$x_1^k+x_2^k+\ldots+x_n^k=1$ for $k=1, 2, \ldots, n-1;$
[*]$x_1^n+x_2^n+\ldots+x_n^n=2;$ and
[*]$x_1^m+x_2^m+\ldots+x_n^m= 4.$
[/list]
Compute the smallest possible value of $m+n.$
Let $x$ and $y$ be two positive real numbers, such that $x + y = 1$. Prove that $$\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9$$
Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.
Let $\omega$ be a circle with positive integer radius $r$. Suppose that it is possible to draw isosceles triangle with integer side lengths inscribed in $\omega$. Compute the number of possible values of $r$ where $1 \le r \le 2023^2$.
Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be $\max \left( 0, 25\left(3 - 2 \max \left( \frac{A}{E} , \frac{E}{A}\right)\right)\right)$, rounded to the nearest integer.
In an acute triangle $ABC$ points $D,E,F$ are located on the sides $BC,CA, AB$ such that \[ \frac{CD}{CE} = \frac{CA}{CB} , \frac{AE}{AF} = \frac{AB}{AC} , \frac{BF}{FD} = \frac{BC}{BA} \] Prove that $AD,BE,CF$ are altitudes of triangle $ABC$.