Found problems: 85335
Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $ X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA+XB+XC$?
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 14\sqrt{3} \qquad
\textbf{(C)}\ \frac{195}{8} \qquad
\textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad
\textbf{(E)}\ \frac{69\sqrt{2}}{4} $
$ C_{1}$ and $ C_{2}$ be two externally tangent circles with diameter $ \left[AB\right]$ and $ \left[BC\right]$, with center $ D$ and $ E$, respectively. Let $ F$ be the intersection point of tangent line from A to $ C_{2}$ and tangent line from $ C$ to $ C_{1}$ (both tangents line on the same side of $ AC$). If $ \left|DB\right|\equal{}\left|BE\right|\equal{}\sqrt{2}$, find the area of triangle $ AFC$.
$\textbf{(A)}\ \frac{7\sqrt{3} }{2} \qquad\textbf{(B)}\ \frac{9\sqrt{2} }{2} \qquad\textbf{(C)}\ 4\sqrt{2} \qquad\textbf{(D)}\ 2\sqrt{3} \qquad\textbf{(E)}\ 2\sqrt{2}$
[b]7.[/b] Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that, with some positive number $C$,
$\sum_{k=1}^{n}k\mid a_k \mid<n C$ ($n=1,2, \dots $)
Putting $s_n= a_0 +a_1+\dots+a_n$, suppose that
$\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s$
exists. Prove that
$\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2$
[b](S. 7)[/b]
If $a,b,c>0$ and $abc=3$,find the biggest value of:
$\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}$
The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the [b]characteristic[/b] of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?
Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$.
Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.
Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?
Let $\omega$ denote the circumscribed circle of an acute-angled $\triangle ABC$ with $AB \neq BC$. Let $A'$ be the point symmetric to the point $A$ with respect to the line $BC$. The lines $AA'$ and $A'C$ intersect $\omega$ for the second time at points $D$ and $E$, respectively. Let the lines $AE$ and $BD$ intersect at point $P$. Prove that the line $A'P$ is tangent to the circumscribed circle of $\triangle A'BC$.
[i]Proposed by Oleksii Masalitin[/i]
For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$.
[i]Proposed by Walther Janous, Austria[/i]
Prove for each non-negative integer $n$ and real number $x$ the inequality \[ \sin{x} \cdot(n \sin{x}-\sin{nx}) \geq 0 \]
Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.
In the triangle $ABC$, $|AB|=15,|BC|=12,|AC|=13$. Let the median $AM$ and bisector $BK$ intersect at point $O$, where $M\in BC,K\in AC$. Let $OL\perp AB,L\in AB$. Prove that $\angle OLK=\angle OLM$.
Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.
Divide the plane into $1$x$1$ squares formed by the lattice points. Let$S$ be the set-theoretic union of a finite number of such cells, and let $a$ be a positive real number less than or equal to 1/4.Show that S can be covered by a finite number of squares satisfying the following three conditions:
1) Each square in the cover is an array of $1$x$1$ cells
2) The squares in the cover have pairwise disjoint interios and
3)For each square $Q$ in the cover the ratio of the area $S \cap Q$ to the area of Q is at least $a$ and at most
$a {(\lfloor a^{-1/2} \rfloor)} ^2$
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector).
At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy?
[i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]
Let $ x$ and $ y$ be positive real numbers with $ x^3 \plus{} y^3 \equal{} x \minus{} y.$ Prove that \[ x^2 \plus{} 4y^2 < 1.\]
Let $X,Y$ be i.i.d $\operatorname{Geom}(p)$. What is the conditional distribution of $X|X+Y=k$?
$\textbf{(A)}~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor\right\}$
$\textbf{(B)}~\operatorname{Uniform}\left\{1,2,\ldots,k\right\}$
$\textbf{(C)}~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor+1\right\}$
$\textbf{(D)}~\text{None of the above}$
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:
[b]I.[/b] $f(1) = 2$
[b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$
Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
Let $ABCD$ be a square, and $C$ the circle whose diameter is $AB.$ Let $Q$ be an arbitrary point on the segment $CD.$ We know that $QA$ meets $C$ on $E$ and $QB$ meets it on $F.$ Also $CF$ and $DE$ intersect in $M.$ show that $M$ belongs to $C.$
Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.
Let $ ABCD$ be a trapezoid with $ AB\parallel CD,AB>CD$ and $ \angle{A} \plus{} \angle{B} \equal{} 90^\circ$. Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.
Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$
for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.
Find all pairs of natural numbers $(a,n)$, $a\geq n \geq 2,$ for which $a^n+a-2$ is a power of $2$.