Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

Triangle $ABC$ is given. Let $O$ be it's circumcenter. Let $I$ be the center of it's incircle.The external angle bisector of $A$ meet $BC$ at $D$. And $I_A$ is the $A$-excenter . The point $K$ is chosen on the line $AI$ such that $AK=2AI$ and $A$ is closer to $K$ than $I$. If the segment $DF$ is the diameter of the circumcircle of triangle $DKI_A$, then prove $OF=3OI$.

Let $ABCD$ be a quadrilateral inscribed in circle of center $O$. The perpendicular on the midpoint $E$ of side $BC$ intersects line $AB$ at point $Z$. The circumscribed circle of the triangle $CEZ$, intersects the side $AB$ for the second time at point $H$ and line $CD$ at point $G$ different than $D$. Line $EG$ intersects line $AD$ at point $K$ and line $CH$ at point $L$. Prove that the points $A,H,L,K$ are concyclic, e.g. lie on the same circle.

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

(a) Three lines $l,m,n$ in space pass through point $S$. A plane perpendicular to $m$ intersects $l,m,n $ at $A,B,C$ respectively. Suppose that $\angle ASB = \angle BSC = 45^o$ and $\angle ABC = 90^o$. Compute $\angle ASC$. (b) Furthermore, if a plane perpendicular to $l$ intersects $l,m,n$ at $P,Q,R$ respectively and $SP = 1$, find the sides of triangle $PQR$.

A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC, BD$ at $M, N$ respectively. Suppose that $MN$ meets $AB,CD$ at $P, Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of $IAB, ICD$ at $K, L$ respectively, which are distinct from $I$. Prove that the lines $PK, QL$, and $OI$ are concurrent. Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh

A convex quadrilateral $ABCD$ has $\angle BAC = \angle ADC$. Let $M{}$ be the midpoint of the diagonal $AC$. The side $AD$ contains a point $E$ such that $ABME$ is a parallelogram. Let $N{}$ be the midpoint of the line segment $AE{}$. Prove that the line $AC$ touches the circumcircle of the triangle $DMN$ at point $M{}$.

$AB$ is a chord of circle $\omega$, $P$ is a point on minor arc $AB$, $E,F$ are on segment $AB$ such that $AE=EF=FB$. $PE,PF$ meets $\omega$ at $C,D$ respectively. Prove that $EF\cdot CD=AC\cdot BD$.

Let $P$ be a point outside a circle $\Gamma$ centered at point $O$, and let $PA$ and $PB$ be tangent lines to circle $\Gamma$. Let segment $PO$ intersect circle $\Gamma$ at $C$. A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$, respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$, compute the area of $\triangle{AOC}$. [i]2020 CCA Math Bonanza Individual Round #6[/i]

Let $M$ be the midpoint of side $AC$ of triangle $ABC$, $MD$ and $ME$ be the perpendiculars from $M$ to $AB$ and $BC$ respectively. Prove that the distance between the circumcenters of triangles $ABE$ and $BCD$ is equal to $AC/4$ [i](Proposed by M.Volchkevich)[/i]

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

The adjacent sides of the decagon shown meet at right angles. What is its perimeter? [asy]defaultpen(linewidth(.8pt)); dotfactor=4; dot(origin);dot((12,0));dot((12,1));dot((9,1));dot((9,7));dot((7,7));dot((7,10));dot((3,10));dot((3,8));dot((0,8)); draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8)--(0,8)--cycle); label("$8$",midpoint(origin--(0,8)),W); label("$2$",midpoint((3,8)--(3,10)),W); label("$12$",midpoint(origin--(12,0)),S);[/asy]$ \textbf{(A)}\ 22\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 50$

[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$? [b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors? [b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$? [b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain? [b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$) [b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$. [b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides? [b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent? [b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$. [b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$? [b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$? [b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon? [b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important? [b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$. [b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/ [b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles? [b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ . [b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer? [b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals? [b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A quadrilateral $ABCD$ is such that $\angle A= \angle C=60^o$ and $\angle B=100^o$. Let $O_1$ and $O_2$ be the centers of the incircles of triangles $ABD$ and $CBD$ respectively. Find the angle between the lines $AO_2$ and $CO_1$.

The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has length [asy] size(250); defaultpen(fontsize(10)); pair A=origin, O=(1,0), B=(2,0), N=(3,0), C=(4,0), P=(5,0), D=(6,0), G=tangent(A,P,1,2), E=intersectionpoints(A--G, Circle(N,1))[0], F=intersectionpoints(A--G, Circle(N,1))[1]; draw(Circle(O,1)^^Circle(N,1)^^Circle(P,1)^^G--A--D, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F^^G^^O^^N^^P); label("$A$", A, W); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, dir(0)); label("$P$", P, S); label("$N$", N, S); label("$O$", O, S); label("$E$", E, dir(120)); label("$F$", F, NE); label("$G$", G, dir(100));[/asy] $\textbf {(A) } 20 \qquad \textbf {(B) } 15\sqrt{2} \qquad \textbf {(C) } 24 \qquad \textbf{(D) } 25 \qquad \textbf {(E) } \text{none of these}$

In a circle of diameter $ 1$, some chords are drawn. The sum of their lengths is greater than $ 19$. Prove that there is a diameter intersecting at least $ 7$ chords.

Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system: \[ \left\{\begin{array}{c}f(x,y)\equal{}0\\ g(x,y)\equal{}0\end{array}\right.\] has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.

Each of the sides $ BC, CA, AB $ of the triangle $ ABC $ was divided into three equal parts and on the middle sections of these sides as bases, equilateral triangles were built outside the triangle $ ABC $, the third vertices of which were marked with the letters $ A', B' , C' $ respectively. In addition, points $ A'', B'', C'' $ were determined, symmetrical to $ A', B', C' $ respectively with respect to the lines $ BC, CA, AB $. Prove that the triangles $ A'B'C' $ and $ A''B''C'' $ are equilateral and have the same center of gravity as the triangle $ ABC $.

Points $A_1$, $A_2$, $B_1$, $B_2$ lie on the circumcircle of a triangle $ABC$ in such a way that $A_1B_1 \parallel AB$, $A_1A_2 \parallel BC$, $B_1B_2 \parallel AC$. The line $AA_2$ and $CA_1$ meet at point $A'$, and the lines $BB_2$ and $CB_1$ meet at point $B'$. Prove that all lines $A'B'$ concur.

In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.

[b]6.1 [/b] Three people with one double seater motorbike simultaneously headed from city A to city B . How should they act so that time, for which the last of them will get to , was the smallest? Determine this time. Pedestrian speed - 5 km/h, motorcycle speed - 45 km/h, distance from A to B is equal to 60 kilometers . [b]6.2 / 7.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$? [b]6.3.[/b] A person's age in $1962$ was one more than the sum of digits of the year of his birth. How old is he? [b]6.4. / 7.3[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area. [b]6.5.[/b] Prove that a $201 \times 201$ chessboard can be bypassed by moving a chess knight, visiting each square exactly once. [b]6.6.[/b] Can an integer whose last two digits are odd be the square of another integer? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

The icosahedron is a convex, regular polyhedron consisting of $20$ equilateral triangle for faces. A particular icosahedron given to you has labels on each of its vertices, edges, and faces. Each minute, you uniformly at random pick one of the labels on the icosahedron. If the label is on a vertex, you remove it. If the label is on an edge, you delete the label on the edge along with any labels still on the vertices of that edge. If the label is on a face, you delete the label on the face along with any labels on the edges and vertices which make up that face. What is the expected number of minutes that pass before you have removed all labels from the icosahedron?