Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

Source: 1976 Euclid Part B Problem 3 ----- $I$ is the centre of the inscribed circle of $\triangle{ABC}$. $AI$ meets the circumcircle of $\triangle{ABC}$ at $D$. Prove that $D$ is equidistant from $I$, $B$, and $C$.

What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle $(a)$ of size $6\times 3$? $(b)$ of size $5\times 3$?

Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.

Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e. an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other.

In a convex quadrilateral $ABCD$ the angles $\angle A$ and $\angle C$ are equal and the bisector of $\angle B$ passes through the midpoint of the side $CD$. If it is known that $CD = 3AD$, calculate $\frac{AB}{BC}$.

A rectangle has been divided into 8 smaller rectangles as shown below. Given the area of seven of these rectangles, find the area of the shaded rectangle. [i]Lightning 2.1[/i]

Let $S$ be a square of side length $1$, one of whose vertices is $A$. Let $S^+$ be the square obtained by rotating S clockwise about $A$ by $30^o$ . Let $S^-$ be the square obtained by rotating S counterclockwise about $A$ by $30^o$. Compute the total area that is covered by exactly two of the squares $S$, $S^+$, $S^-$. Express your answer in the form $a + b\sqrt3$ where $a, b$ are rational numbers.

(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

In the triangle $ABC$, variable points $D, E, F$ are on the sides[lines] $BC, CA, AB$ respectively so the triangle $DFE$ is similar to the triangle $ABC$ in this order. Circumcircles of $BDF$ and $CDE$ intersect respectively the circumcircle of $ABC$ at $P$ and $Q$ for the second time. Prove that the circumcircle of $DPQ$ passes through a fixed point.

Let $\gamma$ and $\Gamma$ be two circles such that $\gamma$ is internally tangent to $\Gamma$ at a point $X$. Let $P$ be a point on the common tangent of $\gamma$ and $\Gamma$ and $Y$ be the point on $\gamma$ other than $X$ such that $PY$ is tangent to $\gamma$ at $Y$. Let $PY$ intersect $\Gamma$ at $A$ and $B$, such that $A$ is in between $P$ and $B$ and let the tangents to $\Gamma$ at $A$ and $B$ intersect at $C$. $CX$ intersects $\Gamma$ again at $Z$ and $ZY$ intersects $\Gamma$ again at $Q$. If $AQ = 6, AB = 10$ and $\tfrac{AX}{XB} = \tfrac{1}{4}$. The length of $QZ = \tfrac{p}{q}\sqrt{r}$ where $p$ and $q$ are coprime positive integers, and $r$ is square free positive integer. Find $p + q + r$.

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA \plus{} < MCB \equal{} MND \plus{} < MBC \equal{} 180^0$. Prove that $ MN$ is parallel to $ AB$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]D1.[/b] What is the solution to the equation $3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6$? [b]D2.[/b] Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make? [b]D3.[/b] What fraction of the multiples of $5$ between $1$ and $100$ inclusive are also multiples of $20$? [b]D4.[/b] What is the maximum number of times a circle can intersect a triangle? [b]D5 / L1.[/b] At an interesting supermarket, the nth apple you purchase costs $n$ dollars, while pears are $3$ dollars each. Given that Layla has exactly enough money to purchase either $k$ apples or $2k$ pears for $k > 0$, how much money does Layla have? [b]D6 / L3.[/b] For how many positive integers $1 \le n \le 10$ does there exist a prime $p$ such that the sum of the digits of $p$ is $n$? [b]D7 / L2.[/b] Real numbers $a, b, c$ are selected uniformly and independently at random between $0$ and $1$. What is the probability that $a \ge b \le c$? [b]D8.[/b] How many ordered pairs of positive integers $(x, y)$ satisfy $lcm(x, y) = 500$? [b]D9 / L4.[/b] There are $50$ dogs in the local animal shelter. Each dog is enemies with at least $2$ other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt. [b]D10 / L7.[/b] Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$. [b]D11 / L8.[/b] The numbers $1$ through $5$ are written on a chalkboard. Every second, Sara erases two numbers $a$ and $b$ such that $a \ge b$ and writes $\sqrt{a^2 - b^2}$ on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair $(M, m)$. [b]D12 / L9.[/b] $N$ people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the $N$ people so that the sum of all the numbers is $1$ and the sum of any three consecutive people’s numbers does not exceed $1/2019$.” If Bella is right, find the minimum value of $N$ possible. [b]D13 / L10.[/b] In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$. [b]D14 / L11.[/b] Consider nonnegative real numbers $a_1, ..., a_6$ such that $a_1 +... + a_6 = 20$. Find the minimum possible value of $$\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.$$ [b]D15 / L13.[/b] Find an $a < 1000000$ so that both $a$ and $101a$ are triangular numbers. (A triangular number is a number that can be written as $1 + 2 +... + n$ for some $n \ge 1$.) Note: There are multiple possible answers to this problem. You only need to find one. [b]L6.[/b] How many ordered pairs of positive integers $(x, y)$, where $x$ is a perfect square and $y$ is a perfect cube, satisfy $lcm(x, y) = 81000000$? [b]L12.[/b] Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the incenter of triangle $ABC$, if it exists. Find the area of the region of points $f(f(X))$ where $X$ is arbitrary. [b]L14.[/b] Leptina and Zandar play a game. At the four corners of a square, the numbers $1, 2, 3$, and $4$ are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers $a$ and $b$ with $a \ge b$ and replace $a$ with $ a - b$. Zandar wants to reduce the sum of the numbers at the four corners of the square to $2$ in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to $2$. [b]L15.[/b] There exist polynomials $P, Q$ and real numbers $c_0, c_1, c_2, ... , c_{10}$ so that the three polynomials $P, Q$, and $$c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}$$ are all polynomials of degree 2019. Suppose that $c_0 = 1$, $c_1 = -7$, $c_2 = 22$. Find all possible values of $c_{10}$. Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

Let \( \triangle ABC \) be an acute-angled triangle. Let \( D \) and \( E \) be the feet of the altitudes from \( B \) and \( C \), respectively. Let \( E' \) be the reflection of point \( E \) with respect to line \( BD \), which is assumed to lie on the circumcircle of triangle \( \triangle ABC \). Let \( C' \) be the reflection of point \( C \) with respect to line \( BD \). Prove that triangle \( C'AE \) is isosceles and determine the ratio \( AD : DC \).

A convex quadrilateral $ ABCD$ is inscribed in a circle. The tangents to the circle through $ A$ and $ C$ intersect at a point $ P$, such that this point $ P$ does not lie on $ BD$, and such that $ PA^{2}=PB\cdot PD$. Prove that the line $ BD$ passes through the midpoint of $ AC$.

Prove that no matter how we number the vertices of a cube with integers from 1 to 8, there exists two opposite vertices in the cube (e.g. they are the endpoints of a large diagonal of the cube), united through a broken line formed with 3 edges of the cube, such that the sum of the 4 numbers written in the vertices of this broken lines is at least 21.

Given is a triangle $\triangle ABC$ and points $M$ and $K$ on lines $AB$ and $CB$ such that $AM=AC=CK$. Prove that the length of the radius of the circumcircle of triangle $\triangle BKM$ is equal to the lenght $OI$, where $O$ and $I$ are centers of the circumcircle and the incircle of $\triangle ABC$, respectively. Also prove that $OI\perp MK$.

$\vartriangle ABC \sim \vartriangle A'B'C'$. $\vartriangle ABC$ has sides $a,b,c$ and $\vartriangle A'B'C'$ has sides $a',b',c'$. Two sides of $\vartriangle ABC$ are equal to sides of $\vartriangle A'B'C'$. Furthermore, $a < a'$, $a < b < c$, $a = 8$. Prove that there is exactly one pair of such triangles with all sides integers.

What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?

The center of the inscribed circle of triangle $ABC$ is $I$. Let $e$ be the perpendicular line on $CI$ passing through $I$. The line $e$ itnersects the side $AC$ at $A'$ and the side $BC$ at point $B'$. Let $A''$ be the symmetric point of $A$ wrt $A'$, $B''$ be the symmetric point of $B$ wrt $B'$. Prove that $A''B''$ is a line tangent to the incircle.

Problem: A mathematical moron is given the values b; c; A for a triangle ABC and is required to find a. He does this by using the cosine rule $ a^2 = b^2 + c^2 - 2bccosA$ and misapplying the low of the logarithm to this to get $ log a^2 = log b^2 + log c^2 - log(2bc cos A) $ He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.