Given is a triangle $ABC$ with altitude $AH$, diameter of the circumcircle $AD$ and incenter $I$. Prove that $\angle BIH = \angle DIC$.

In an acute-angled triangle $ ABC $ consider $ A_1,B_1,C_1 $ to be the symmetric points of the orthocenter of $ ABC $ to the sides $ BC,AC,AB, $ respectively. Show that if the centroids of the triangles $ ABC,A_1B_1C_1 $ are the same, then $ ABC $ is equilateral. [i]Carmen Botea[/i]

All the sides of the convex hexagon $ABCDEF$ are equal. In addition, $AD = BE = CF$. Prove that a circle can be inscribed into this hexagon. [i](Boyan Obukhov)[/i]

Let $p, q$ and $r$ be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes $\alpha, \beta$, and $\gamma$, containing $p, q$, and $r$ respectively, that are perpendicular to each other $(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).$

Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively. A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$ Prove that: $ X,\,Y,\,Z$ are collinear.

Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.

Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly. (a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel. (b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear. [i]Nikolai Beluhov[/i]

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

In a triangle $\triangle{ABC}$ with all its sides of different length, $D$ is on the side $AC$, such that $BD$ is the angle bisector of $\sphericalangle{ABC}$. Let $E$ and $F$, respectively, be the feet of the perpendicular drawn from $A$ and $C$ to the line $BD$ and let $M$ be the point on $BC$ such that $DM$ is perpendicular to $BC$. Show that $\sphericalangle{EMD}=\sphericalangle{DMF}$.

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$

P6. It is given the integer $Y$ with $Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$ Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.) P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is .... P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$. P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal. (Image should be placed here, look at attachment.) a) Determine the position of the number $2018$ based on its row, column, and diagonal. b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$. P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a [i]gadang[/i] number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a [i]gadang[/i] number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a [i]gadang[/i] number?

Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$ and $ \angle A\equal{}20^\circ$. On the side $ AC$ consider point $ D$ such that $ AD\equal{}BC$. Find $ \angle BDC$.

$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that \[\cos B+\cos C=1\Longleftrightarrow MI=MT\]

Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

Let $ABC$ be an equilateral triangle. Suppose $D$ is the point on $BC$ such that $BD+DC = 1:3$. Let the perpendicular bisector of $AD$ intersect $AB,AC$ at $E,F$ respectively. Prove that $49 \cdot [BDE] = 25 \cdot [CDF]$, where $[XYZ]$ denotes the area of the triangle $XYZ$.

Let $ABCD$ be a rectangle and $U, V$ two points of its circumcircle. Lines $AU,CV$ intersect at $P$ and lines $BU,DV$ intersect at $Q$, distinct from $P$. Prove that $$\frac{1}{PQ^2} \ge \frac{1}{UV^2} - \frac{1}{AC^2}$$ Michel Bataille

Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ externally, the arc $BD$ externally and also touching the side $AD$. Find the radius of $\Gamma$.

[u]Round 1[/u] [b]1.1.[/b] There are $99$ dogs sitting in a long line. Starting with the third dog in the line, if every third dog barks three times, and all the other dogs each bark once, how many barks are there in total? [b]1.2.[/b] Indigo notices that when she uses her lucky pencil, her test scores are always $66 \frac23 \%$ higher than when she uses normal pencils. What percent lower is her test score when using a normal pencil than her test score when using her lucky pencil? [b]1.3.[/b] Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.) [u]Round 2[/u] [b]2.1.[/b] If $x + 3y = 2$, then what is the value of the expression $9^x * 729^y$? [b]2.2.[/b] Lazy Sheep loves sleeping in, but unfortunately, he has school two days a week. If Lazy Sheep wakes up each day before school's starting time with probability $1/8$ independent of previous days, then the probability that Lazy Sheep wakes up late on at least one school day over a given week is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]2.3.[/b] An integer $n$ leaves remainder $1$ when divided by $4$. Find the sum of the possible remainders $n$ leaves when divided by $20$. [u]Round 3[/u] [b]3.1. [/b]Jake has a circular knob with three settings that can freely rotate. Each minute, he rotates the knob $120^o$ clockwise or counterclockwise at random. The probability that the knob is back in its original state after $4$ minutes is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]3.2.[/b] Given that $3$ not necessarily distinct primes $p, q, r$ satisfy $p+6q +2r = 60$, find the sum of all possible values of $p + q + r$. [b]3.3.[/b] Dexter's favorite number is the positive integer $x$, If $15x$ has an even number of proper divisors, what is the smallest possible value of $x$? (Note: A proper divisor of a positive integer is a divisor other than itself.) [u]Round 4[/u] [b]4.1.[/b] Three circles of radius $1$ are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-\sqrt{c}}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relatively prime. Find $a + b + c + d$. [b]4.2. [/b]Evaluate $$\frac{\sqrt{15}}{3} \cdot \frac{\sqrt{35}}{5} \cdot \frac{\sqrt{63}}{7}... \cdot \frac{\sqrt{5475}}{73}$$ [b]4.3.[/b] For any positive integer $n$, let $f(n)$ denote the number of digits in its base $10$ representation, and let $g(n)$ denote the number of digits in its base $4$ representation. For how many $n$ is $g(n)$ an integer multiple of $f(n)$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784571p24468619]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$

The lengths of sides $CB$ and $CA$ of $\vartriangle ABC$ are $a$ and $b$, and the angle between them is $\gamma = 120^o$. Express the length of the bisector of $\gamma$ in terms of $a$ and $b$.

Let $D$ and $E$ be points on $[BC]$ and $[AC]$ of acute $\triangle ABC$, respectively. $AD$ and $BE$ meet at $F$. If $|AF|=|CD|=2|BF|=2|CE|$, and $Area(\triangle ABF) = Area(\triangle DEC)$, then $Area(\triangle AFC)/Area(\triangle BFC) = ?$ $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt2 \qquad \textbf{(E)}\ 1$