Found problems: 1679
A point $D$ lies inside triangle $ABC$. Let $A_1, B_1, C_1$ be the second intersection points of the lines $AD$, $BD$, and $CD$ with the circumcircles of $BDC$, $CDA$, and $ADB$, respectively. Prove that
$$\frac{AD}{AA_1} + \frac{BD}{BA_1} + \frac{CD}{CC_1} = 1.$$
A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.
The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$.
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On the side $AB$ of the triangle $ABC$ mark the point $K$. The segment $CK$ intersects the median $AM$ at the point $F$. It is known that $AK = AF$. Find the ratio $MF: BK$.
Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?
Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 3 \sqrt{3} \qquad
\textbf{(C)}\ 3 \sqrt{6} \qquad
\textbf{(D)}\ 6 \sqrt{3} \qquad
\textbf{(E)}\ 6 \sqrt{6}$
(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$
(2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$
Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.
Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
Let $ A_1,B_1,C_1 $ be the feet of the heights of an acute triangle $ ABC. $ On the segments $ B_1C_1,C_1A_1,A_1B_1, $ take the points $ X,Y, $ respectively, $ Z, $ such that
$$ \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\ \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . $$
Show that $ AX,BY,CZ, $ are concurrent.
If the ratio of the legs of a right triangle is $ 1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
$ \textbf{(A)}\ 1: 4 \qquad
\textbf{(B)}\ 1: \sqrt{2} \qquad
\textbf{(C)}\ 1: 2 \qquad
\textbf{(D)}\ 1: \sqrt{5} \qquad
\textbf{(E)}\ 1: 5$
Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.
The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is
$\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$
In quadrilateral $ ABCD$, it is given that $ \angle A \equal{} 120^\circ$, angles $ B$ and $ D$ are right angles, $ AB \equal{} 13$, and $ AD \equal{} 46$. Then $ AC \equal{}$
$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 72$
Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series.
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
$ \textbf{(A) } \frac 67 \qquad \textbf{(B) } \frac {13}{14} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \frac {14}{13} \qquad \textbf{(E) } \frac 76$