Found problems: 1581
Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i]
On a plane, given a square $D$ with side length 1 and a line which intersects with $D$. For the solid obtained by a rotation of $D$ about the line as the axis, answer the following questions:
(1) Suppose that the line $l$ on a plane the same with $D$ isn't parallel to any edges. Prove that the line by which the volume of the solid is maximized has only intersection point with $D$. Note that the line as axis of rotation is parallel to $l$.
(2) Find the possible maximum volume for which all solid formed by the rotation axis as line intersecting with $D$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 4[/i]
In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.
Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.
Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?
Consider the graphs of (1): $ y\equal{}x^2\minus{}\frac{1}{2}x\plus{}2$ and (2) $ y\equal{}x^2\plus{}\frac{1}{2}x\plus{}2$ on the same set of axis. These parabolas are exactly the same shape. Then:
$ \textbf{(A)}\ \text{the graphs coincide.} \\
\textbf{(B)}\ \text{the graph of (1) is lower than the graph of (2).} \\
\textbf{(C)}\ \text{the graph of (1) is to the left of the graph of (2).} \\
\textbf{(D)}\ \text{the graph of (1) is to the right of the graph of (2).} \\
\textbf{(E)}\ \text{the graph of (1) is higher than the graph of (2).}$
The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively.
Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.
$ ABCD$ is a square of side length $ \sqrt{3} + 1 $. Point $ P $ is on $ \overline{AC} $ such that $ AP = \sqrt{2} $. The square region bounded by $ ABCD $ is rotated $ 90^{\circ} $ counterclockwise with center $ P $, sweeping out a region whose area is $ \frac{1}{c} (a \pi + b) $, where $a $, $b$, and $ c $ are positive integers and $ \text{gcd}(a,b,c) = 1 $. What is $ a + b + c $?
$\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23 $
Let $AB$ be a diameter of a semicircle $\Gamma$. Two circles, $\omega_1$ and $\omega_2$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $AB$ at $P$ and $Q$, respectively, and to semicircular arc $AB$ at $C$ and $D$, respectively, with $AP<AQ$. Suppose $F$ lies on $\Gamma$ such that $ \angle FQB = \angle CQA $ and that $ \angle ABF = 80^\circ $. Find $ \angle PDQ $ in degrees.
In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.
Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity.
(a) Determine the locus of the midpoint of $ M_1M_2$.
(b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.
Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$.
$Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$.
$Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$.
Determine $\frac{|PB|}{|AB|}$ if $S=S'$.
$M$ is a finite set of points in a plane. Point $O$ in the plane is called an “almost centre of symmetry” of set $M$ if it is possible to remove from $M$ one point in such a way that among the remaining members $O$ is the centre of symmetry in the usual sense. How many such “almost centres of symmetry” may a finite point set in a plane have? Indicate all such points.
(V Prasolov, Moscow)
The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is
A. $ a\plus{}b\plus{}c$
B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$
C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$
D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$
E. None of these
In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.
(Here we denote $XY$ the length of the line segment $XY$.)
Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that
\[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]
Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color.
[b]
proposed by S. Berlov[/b]
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.
Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.
For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and
let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows.
(1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$.
(2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$
(3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.
Let $ABC$ be a triangle. Let $C^{\prime}$ and $A^{\prime}$ be the reflections of its vertices $C$ and $A$, respectively, in the altitude of triangle $ABC$ issuing from $B$. The perpendicular to the line $BA^{\prime}$ through the point $C^{\prime}$ intersects the line $BC$ at $U$; the perpendicular to the line $BC^{\prime}$ through the point $A^{\prime}$ intersects the line $BA$ at $V$. Prove that $UV \parallel CA$.
Darij
The integers from $1$ to $9$ can be arranged into a $3\times3$ array (as shown on the right) so that the sum of the numbers in every row, column, and diagonal is a multiple of $9$.
(a.) Prove that the number in the center of the array must be a multiple of $3$.
(b.) Give an example of such an array with $6$ in the center.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));size(100);
int i,j;
for(i=0; i<4; i=i+1) {
draw((0,2i)--(6,2i));
draw((2i,0)--(2i,6));
}
string[] letters={"G", "H", "I", "D", "E", "F", "A", "B", "C"};
for(i=0; i<3; i=i+1) {
for(j=0; j<3; j=j+1) {
label(letters[3i+j], (2j+1, 2i+1));
}}[/asy]
Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form
\[ 12x^2 + bxy + cy^2 + d = 0. \]
Find the product $bc$.