Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.

The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$

In triangle $ABC$, $AC=7$. $D$ lies on $AB$ such that $AD=BD=CD=5$. Find $BC$.

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid? $ \textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt{6} \qquad \textbf{(C)}\ 392\sqrt2 \qquad \textbf{(D)}\ 392\sqrt3 \qquad \textbf{(E)}\ 784$

Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?

$ABC$ is acute-angled triangle. $AA_1,BB_1,CC_1$ are altitudes. $X,Y$ - midpoints of $AC_1,A_1C$. $XY=BB_1$. Prove that one side of $ABC$ in $\sqrt{2}$ greater than other side.

In the acute-angled triangle $ABC$, let $AP$ and $BQ$ be the altitudes, $CM$ be the median . Point $R$ is the midpoint of $CM$. Line $PQ$ intersects line $AB$ at $T$. Prove that $OR \perp TC$, where $O$ is the center of the circumscribed circle of triangle $ABC$.

[b]p1.[/b] In rectangle $ABMC$, $AB= 5$ and $BM= 8$. If point $X$ is the midpoint of side $AC$, what is the area of triangle $XCM$? [b]p2.[/b] Find the sum of all possible values of $a+b+c+d$ such that $(a, b, c, d)$ are quadruplets of (not necessarily distinct) prime numbers satisfying $a \cdot b \cdot c \cdot d = 4792$. [b]p3.[/b] How many integers from $1$ to $2022$ inclusive are divisible by $6$ or $24$, but not by both? [b]p4.[/b] Jerry begins his English homework at $07:39$ a.m. At $07:44$ a.m., he has finished $2.5\%$ of his homework. Subsequently, for every five minutes that pass, he completes three times as much homework as he did in the previous five minute interval. If Jerry finishes his homework at $AB : CD$ a.m., what is $A + B + C + D$? For example, if he finishes at $03:14$ a.m., $A + B + C + D = 0 + 3 + 1 + 4$. [b]p5.[/b] Advay the frog jumps $10$ times on Mondays, Wednesdays and Fridays. He jumps $7$ times on Tuesdays and Saturdays. He jumps $5$ times on Thursdays and Sundays. How many times in total did Advay jump in November if November $17$th falls on a Thursday? (There are $30$ days in November). [b]p6.[/b] In the following diagram, $\angle BAD\cong \angle DAC$, $\overline{CD} = 2\overline{BD}$, and $ \angle AEC$ and $\angle ACE$ are complementary. Given that $\overline{BA} = 210$ and $\overline{EC} = 525$, find $\overline{AE}$. [img]https://cdn.artofproblemsolving.com/attachments/5/3/8e11caf2d7dbb143a296573f265e696b4ab27e.png[/img] [b]p7.[/b] How many trailing zeros are there when $2021!$ is expressed in base $2021$? [b]p8.[/b] When two circular rings of diameter $12$ on the Olympic Games Logo intersect, they meet at two points, creating a $60^o$ arc on each circle. If four such intersections exist on the logo, and no region is in $3$ circles, the area of the regions of the logo that exist in exactly two circles is $a\pi - b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $\sqrt{c}$ is fully simplified find $a + b + c$. [b]p9.[/b] If $x^2 + ax - 3$ is a factor of $x^4 - x^3 + bx^2 - 5x - 3$, then what is $|a + b|$? [b]p10.[/b] Let $(x, y, z)$ be the point on the graph of $x^4 +2x^2y^2 +y^4 -2x^2 -2y^2 +z^2 +1 = 0$ such that $x+y +z$ is maximized. Find $a+b$ if $xy +xz +yz$ can be expressed as $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers. [b]p11.[/b] Andy starts driving from Pittsburgh to Columbus and back at a random time from $12$ pm to $3$ pm. Brendan starts driving from Pittsburgh to Columbus and back at a random time from $1$ pm to $4$ pm. Both Andy and Brendan take $3$ hours for the round trip, and they travel at constant speeds. The probability that they pass each other closer to Pittsburgh than Columbus is$ m/n$, for relatively prime positive integers $m$ and $n$. What is $m + n$? [b]p12.[/b] Consider trapezoid $ABCD$ with $AB$ parallel to $CD$ and $AB < CD$. Let $AD \cap BC = O$, $BO = 5$, and $BC = 11$. Drop perpendicular $AH$ and $BI$ onto $CD$. Given that $AH : AD = \frac23$ and $BI : BC = \frac56$ , calculate $a + b + c + d - e$ if $AB + CD$ can be expressed as $\frac{a\sqrt{b} + c\sqrt{d}}{e}$ where $a$, $b$, $c$, $d$, $e$ are integers with $gcd(a, c, e) = 1$ and $\sqrt{b}$, $\sqrt{d}$ are fully simplified. [b]p13.[/b] The polynomials $p(x)$ and $q(x)$ are of the same degree and have the same set of integer coefficients but the order of the coefficients is different. What is the smallest possible positive difference between $p(2021)$ and $q(2021)$? [b]p14.[/b] Let $ABCD$ be a square with side length $12$, and $P$ be a point inside $ABCD$. Let line $AP$ intersect $DC$ at $E$. Let line $DE$ intersect the circumcircle of $ADP$ at $F \ne D$. Given that line $EB$ is tangent to the circumcircle of $ABP$ at $B$, and $FD = 8$, find $m + n$ if $AP$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. [b]p15.[/b] A three digit number $m$ is chosen such that its hundreds digit is the sum of the tens and units digits. What is the smallest positive integer $n$ such that $n$ cannot divide $m$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.

Circle with center $I$, inscribed in a triangle $ABC$ , touches the sides $BC$ and $AC$ at points $A_1$ and $B_1$ respectively. On rays $A_1I$ and $B_1I$, respectively, let be the points $A_2$ and $B_2$ such that $IA_2=IB_2=R$, where $R$is the radius of the circumscribed circle of the triangle $ABC$. Prove that: a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$, b) lines $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$.

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

In an acute-angled triangle $ABC$, $M$ is a point on the side $BC$, the line $AM$ meets the circumcircle $\omega$ of $ABC$ at the point $Q$ distinct from $A$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to the diameter $AK$ of $\omega$ at the point $P$. Let $L$ be the point on $\omega$ distinct from $Q$ such that $PL$ is tangent to $\omega$ at $L$. Prove that $L,M$ and $K$ are collinear.

On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?

The tangent to the circle circumscribed to the triangle $ABC$, taken through the vertex $A$, intersects the line $BC$ at the point $P$, and the tangents to the same circle, taken through $B$ and $C$, intersect the lines $AC$ and $AB$, respectively at the points $Q$ and $R$. Prove that the points $P, Q$ ¸ and $R$ are collinear.

In $\bigtriangleup$$ABC$ $BL$ is bisector. Arbitrary point $M$ on segment $CL$ is chosen. Tangent to $\odot$$(ABC)$ at $B$ intersects $CA$ at $P$. Tangents to $\odot$$BLM$ at $B$ and $M$ intersect at point $Q$. Prove that $PQ$$\parallel$$BL$.

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$? $\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$

Let $ABC$ be a triangle satisfying $AB<AC$. Let $M$ be the midpoint of $BC$. A point $P$ lies on the segment $AB$ with $AP>PB$. A point $Q$ lies on the segment $AC$ with $\angle MPA = \angle AQM$. The perpendicular bisectors of $BC$ and $PQ$ intersect at $S$. Prove that $\angle BAC + \angle QSP = \angle QMP$.

The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$. Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$. Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .

Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.