In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P\ne B$, and circle $CHD$ meets $AC$ at $Q\ne C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle. [i]Michael Ren[/i]

Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?

Robin has obtained a circular pizza with radius $2$. However, being rebellious, instead of slicing the pizza radially, he decides to slice the pizza into $4$ strips of equal width both vertically and horizontally. What is the area of the smallest piece of pizza?

Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE \equal{} \angle ABC$.

The satellites $A$ and $B$ circle the Earth in the equatorial plane at altitude $h$. They are separated by distance $2r$, where $r$ is the radius of the Earth. For which $h$ can they be seen in mutually perpendicular directions from some point on the equator?

Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt3$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{m\pi}{n}$ forrelatively prime positive integers $m, n$, find $m + n$. (Two circles intersect orthogonally at a point $P$ if the tangent lines at $P$ form a right angle.)

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

$10.$ A $2\times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square $?$

Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$. Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.) [i]Luxembourg (Pierre Haas)[/i]

Let $ABC$ be a triangle with $m(\widehat A) = 70^\circ$ and the incenter $I$. If $|BC|=|AC|+|AI|$, then what is $m(\widehat B)$? $ \textbf{(A)}\ 35^\circ \qquad\textbf{(B)}\ 36^\circ \qquad\textbf{(C)}\ 42^\circ \qquad\textbf{(D)}\ 45^\circ \qquad\textbf{(E)}\ \text{None of above} $

Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

A triangle $\vartriangle ABC$ has $\angle A = 90^o$, and a point $D$ is chosen on $AC$. Point $F$ is the foot of altitude from $A$ to $BC$. Suppose that $BD = DC = CF = 2$. Compute $AC$.

Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=|AC|$, $|BD|=6$ and $|DC|=10$. If the incircles of $\triangle ABD$ and $\triangle ADC$ touch side $[AD]$ at $E$ and $F$, respectively, what is$|EF|$? $ \textbf{(A)}\ \dfrac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \dfrac{2}{\sqrt{3}} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \dfrac{9}{8} \qquad\textbf{(E)}\ 2 $

The perpendicular bisector of the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$ Show that if there exists a right circular cone with vertex $V$, with the properties: [b](1)[/b] its axis passes through $O$, and [b](2)[/b] its curved surface passes through $A,B,C$ and $D,$ then \[OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.\] Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties [b](1) [/b]and [b](2).[/b]

Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at $A$, $B$, and $C$. Compute the area of triangle $ABC$. [i]2021 CCA Math Bonanza Team Round #6[/i]

Let $ABC$ be a triangle with altitude $\overline{AF}.$ Let $AB=5, AC=8, BC=7.$ Let $P$ be on $\overline{AF}$ such that it lies between $A$ and $F.$ Let $\omega_1, \omega_2$ be the circumcircles of $APB, APC$ respectively. Let $\overline{BC}$ intersect $\omega_1$ at $B' \neq B.$ Also, let $\overline{BC}$ intersect $\omega_2$ at $C' \neq C.$ Let $X \neq A$ be on $\omega_1$ such that $B'X=B'A.$ Let $Y \neq A$ be on $\omega_2$ such that $C'A=C'Y.$ Let $X, Y, A$ all lie on one line $h.$ Find the length of $PA.$

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively. (a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic. (b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.

Karl's rectangular vegetable garden is $20$ by $45$ feet, and Makenna's is $25$ by $40$ feet. Which garden is larger in area? $\textbf{(A)}$ Karl's garden is larger by 100 square feet. $\textbf{(B)}$ Karl's garden is larger by 25 square feet. $\textbf{(C)}$ The gardens are the same size. $\textbf{(D)}$ Makenna's garden is larger by 25 square feet. $\textbf{(E)}$ Makenna's garden is larger by 100 square feet.

$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $ AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90$. Prove that $\angle XMN=\angle ABC$

[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1[/url] The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$ [b]8.2[/b] In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle. [url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3[/url] Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities? [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4*[/url] (asterisk problems in separate posts) [url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5[/url] Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers. [url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6[/url] Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.