grade IX P4, X P3 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the circumscirbed circle of this triangle at the points $ A_0 $ and $ C_0 $, respectively. The straight line passing through the center of the inscribed circle of triangle $ ABC $ parallel to the side of $ AC $, intersects with the line $ A_0C_0 $ at $ P $. Prove that the line $ PB $ is tangent to the circumcircle of the triangle $ ABC $. grade XI P4 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the sides at the points $ A_1 $ and $ C_1 $, and the circumcircle of this triangle at points $ A_0 $ and $ C_0 $ respectively. Straight lines $ A_1C_1 $ and $ A_0C_0 $ intersect at point $ P $. Prove that the segment connecting $ P $ with the center inscribed circles of triangle $ ABC $, parallel to $ AC $.

Let $ABC$ be an acute scalene triangle such that $AB <AC$. The midpoints of sides $AB$ and $AC$ are $M$ and $N$, respectively. Let $P$ and $Q$ be points on the line $MN$ such that $\angle CBP = \angle ACB$ and $\angle QCB = \angle CBA$. The circumscribed circle of triangle $ABP$ intersects line $AC$ at $D$ ($D\ne A$) and the circumscribed circle of triangle $AQC$ intersects line $AB$ at $E$ ($E \ne A$). Show that lines $BC, DP,$ and $EQ$ are concurrent. Nicolás De la Hoz, Colombia

In convex pentagon $ ABCDE$, denote by $ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command. = B',CF%Error. "capDB" is a bad command. = C',DG\cap EC = D',EH\cap AD = E'.$ Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.

Given a regular convex $2m$- sided polygon $P$, show that there is a $2m$-sided polygon $\pi$ with the same vertices as $P$ (but in different order) such that $\pi$ has exactly one pair of parallel sides.

Three equal circles of radius $r$ each pass through the centres of the other two. What is the area of intersection that is common to all the three circles?

Let $ABC$ be an acute traingle with $AC \neq BC$. Point $K$ is a foot of altitude through vertex $C$. Point $O$ is a circumcenter of $ABC$. Prove that areas of quadrilaterals $AKOC$ and $BKOC$ are equal.

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(0));[/asy]

[b]p1.[/b] Function $f$ is defined by $f (x) = ax+b$ for some real values $a, b > 0$. If $f (f (x)) = 9x + 5$ for all $x$, find $b$. [b]p2.[/b] At some point during a game, Will Avery has made $1/3$ of his shots. When he shoots once and makes a basket, his average increases to $2/5$. Find his average (expressed as a fraction) after a second additional basket. [b]p3.[/b] A dealer has a deck of $1999$ cards. He takes the top card off and “ducks” it, that is, places it on the bottom of the deck. He deals the second card onto the table. He ducks the third card, deals the fourth card, ducks the fifth card, deals the sixth card, and so forth, continuing until he has only one card left; he then ducks the last card with itself and deals it. Some of the cards (like the second and fourth cards) are not ducked at all before being dealt, while others are ducked multiple times. The question is: what is the average number of ducks per card? [b]p4.[/b] Point $P$ lies outside circle $O$. Perpendicular lines $\ell$ and m intersect at $P$. Line $\ell$ is tangent to circle $O$ at a point $6$ units from $P$. Line $m$ crosses circle $O$ at a point $4$ units from $P$. Find the radius of circle $O$. [b]p5.[/b] Define $f(n)$ by $$f(n) = \begin{cases} n/2 \,\,\,\text{if} \,\,\, n\,\,\,is\,\,\, even \\ (n + 1023)/2\,\,\, \text{if} \,\,\, n\,\,\,is\,\,\, odd \end{cases}$$ Find the least positive integer $n$ such that $f(f(f(f(f(n))))) = n.$ [b]p6.[/b] Write $\sqrt{10001}$ to the sixth decimal place, rounding down. [b]p7.[/b] Define $(a_n)$ recursively by $a_1 = 1$, $a_n = 20 \cos (a_{n-1}^o)$. As $n$ tends to infinity, $(a_n)$ tends to $18.9195...$. Define $(b_n)$ recursively by $b_1 = 1$, $b_n =\sqrt{800 + 800 \cos (b_{n-1}^o)}$. As $n$ tends to infinity, $(b_n)$ tends to $x$. Calculate $x$ to three decimal places. [b]p8.[/b] Let $mod_d (k)$ be the remainder of $k$ when divided by $d$. Find the number of positive integers $n$ satisfying $$mod_n(1999) = n^2 - 89n + 1999$$ [b]p9.[/b] Let $f(x) = x^3 + x$. Compute $$\sum^{10}_{k=1} \frac{1}{1 + f^{-1}(k - 1)^2 + f^{-1}(k - 1)f^{-1}(k) + f^{-1}(k)^2}$$ ($f^{-1}$ is the inverse of $f$: $f (f^{-1}1 (x)) = f^{-1}1 (f (x)) = x$ for all $x$.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/d/6/5e3faa5755628cceb2b5c39c95f6126669a3c6.png[/img]

A real number $f(X)\neq 0$ is assigned to each point $X$ in the space. It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : \[ f(O)=f(A)f(B)f(C)f(D). \] Prove that $f(X)=1$ for all points $X$. [i]Proposed by Aleksandar Ivanov[/i]

Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions. (1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say,　the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$.　Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$.. (2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$. (3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.

Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

Given a polygon $ABCDEFGHIJ$. How many diagonals does the polygon have?

Let $ABC$ be a triangle. It is known that the triangle formed by the midpoints of the medians of $ABC$ is equilateral. Prove that $ABC$ is equilateral as well.

Let $[ABC]$ be any triangle and let $D, E$ and $F$ be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles $[ABC]$ and $[DEF]$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/6/45bd929dfff87fb8deb09eddb59ef46e0dc0f4.png[/img]

Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.

Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.

Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

Consider a $2n \times 2n$ board. From the $i$th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board?

In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it. (A. Zaslavsky)

Distinct points $B,B',C,C'$ lie on an arbitrary line $\ell$. $A$ is a point not lying on $\ell$. A line passing through $B$ and parallel to $AB'$ intersects with $AC$ in $E$ and a line passing through $C$ and parallel to $AC'$ intersects with $AB$ in $F$. Let $X$ be the intersection point of the circumcircles of $\triangle{ABC}$ and $\triangle{AB'C'}$($A \neq X$). Prove that $EF \parallel AX$.

If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is: $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{3}{4} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{5}$

Consider the parallelogram $ABCD$, with $\angle ABC = 60$ and sides $AB =\sqrt3$, $BC = 1$. Let $\omega$ be the circle of center $B$ and radius $BA$, and let $\tau$ be the circle of center $D$ and radius $DA$. Determine the area of the region between the circumferences $\omega$ and $\tau$, within the parallelogram $ABCD$ (the area of the shaded region). [img]https://cdn.artofproblemsolving.com/attachments/5/a/02b17ec644289d95b6fce78cb5f1ecb3d3ba5b.png[/img]

Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.