The incircle of $\vartriangle ABC$ is tangent to sides $\overline{BC}, \overline{AC}$, and $\overline{AB}$ at $D, E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$ as shown. The sides of $\vartriangle ABC$ have lengths $AB = 73, BC = 123$, and $AC = 120$. Find the length $EG$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/aede28071a1a6b94bbe3ad8e1e104822b89439.png[/img]

In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that: $ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$

Let’s consider three pairwise non-parallel straight constant lines in the plane. Three points are moving along these lines with different nonzero velocities, one on each line (we consider the movement to have taken place for infinite time and continue infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some “zero” moment in such a way that the points never were, are or will be collinear?

Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$, and $M$ are collinear. [i]Proposed by Nikola Velov, North Macedonia[/i]

There are three different points on the line $ A $, $ B $, $ C $ and a point $ S $ outside this line; perpendicularly drawn at points $ A $, $ B $, $ C $ to the lines $ SA $, $ SB $, $ SC $ intersect at points $ M $, $ N $, $ P $. Prove that the points $ M $, $ N $, $ P $, $ S $ lie on the circle

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$

[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$. [b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once. [b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle. [b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself. [b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe? [b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$? [b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$. [b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers. [b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days. [b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it. [b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$. [b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places. [b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect. [b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$. [b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ . [b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$. [b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$? [b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$. [b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$. [b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ . [b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$. [b]p23.[/b] Let $a,b,c$ be positive real numbers such that (i) $c > a$ (ii) $10c = 7a +4b +2024$ (iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$. Find $a +b +c$. [b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$. [b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

In an acute triangle $ABC$ let $AD$ and $BE$ be altitudes, and $AP$ and $BQ$ be bisectors. Let $I$ and $O$ be centers of incircle and circumcircle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear.

Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Clyde is making a Pacman sticker to put on his laptop. A Pacman sticker is a circular sticker of radius $3$ inches with a sector of $120^o$ cut out. What is the perimeter of the Pacman sticker in inches?

In the following figure---not drawn to scale!---$E$ is the midpoint of $BC$, triangle $FEC$ has area $7$, and quadrilateral $DBEG$ has area $27$. Triangles $ADG$ and $GEF$ have the same area, $x$. Find $x$. [asy] unitsize(2cm); pair A = (0,38/16); pair B = (0,0); pair C = (38/16,0); pair D = (0,25/16); pair E = (19/16,0); pair F = .4*D+.6*C; draw(D -- C -- B -- A -- E -- F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, S); label("$D$", D, W); label("$E$", E, S); label("$F$", F, N); label("$G$", (17*F-8*C)/9, NE); [/asy]

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

Find the area of a triangle with angles $\frac{1}{7} \pi$, $\frac{2}{7} \pi$, and $\frac{4}{7} \pi $, and radius of its circumscribed circle $R=1$.

Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$. $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$. Let $D$ be the extouch at $BC$. Show that $AD$ is perpendicular to $B_2C_2$

Let $ABCD$ be a convex quadrilateral with $m(\widehat{ADC}) = 90^{\circ}$. The line through $D$ which is parallel to $BC$ meets $AB$ at $E$. If $m(\widehat{DAC}) = m(\widehat{DAE})$, $|AB|=3$ and $|AC|=4$, then $|AE| = ?$ $\textbf{(A)}\ \frac56 \qquad\textbf{(B)}\ \frac13 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \frac34$

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)

The extensions of sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ intersect at $K$. It is known that $AD = BC$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$. Prove that the triangle $MNK$ is obtuse. [i](5 points)[/i]

Given is a trapezoid $ABCD$, $AD \parallel BC$. The angle bisectors of the two pairs of opposite angles meet at $X, Y$. Prove that $AXYD$ and $BXYC$ are cyclic.

Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.

Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.

Let $ABC$ a triangle and $C$ it´s circuncircle. Let $D$ a point in arc $AB$ that not contain $A$, diferent of $B$ and $C$ such that $CD$ and $AB$ are not parallel. Let $E$ the intersection of $CD$ and $AB$ and $O$ the circumcircle of triangle $DBE$. Prove that the measure of $\angle OBE$ does not depend of the choice of $D$.

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

Let $ABCD$ be a trapezoid with bases $ AB$ and $CD$ $(AB>CD)$. Diagonals $AC$ and $BD$ intersect in point $ N$ and lines $AD$ and $BC$ intersect in point $ M$. The circumscribed circles of $ADN$ and $BCN$ intersect in point $ P$, different from point $ N$. Prove that the angles $AMP$ and $BMN$ are equal.