The diagram below shows a $1\times2\times10$ duct with $2\times2\times2$ cubes attached to each end. The resulting object is empty, but the entire surface is solid sheet metal. A spider walks along the inside of the duct between the two marked corners. There are positive integers $m$ and $n$ so that the shortest path the spider could take has length $\sqrt{m}+\sqrt{n}$. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(origin--(43,0)--(61,20)--(18,20)--cycle--(0,-43)--(43,-43)--(43,0)^^(43,-43)--(61,-23)--(61,20)); draw((43,-43)--(133,57)--(90,57)--extension((90,57),(0,-43),(61,20),(18,20))); draw((0,-43)--(0,-65)--(43,-65)--(43,-43)^^(43,-65)--(133,35)--(133,57)); draw((133,35)--(133,5)--(119.5,-10)--(119.5,20)^^(119.5,-10)--extension((119.5,-10),(100,-10),(43,-65),(133,35))); dot(origin^^(133,5)); [/asy]

Let $\vartriangle ABC$ be an acute triangle. $R$ is a point on arc $BC$. Choose two points $P, Q$ on $AR$ such that $B,P,C,Q$ are concyclic. Let the second intersection of $BP$, $CP$, $BQ$, $CQ$ and the circumcircle of $\vartriangle ABC$ is $P_B$, $P_C$, $Q_B$, $Q_C$, respectively. Let the circumcenter of $\vartriangle P P_BP_C$ and $\vartriangle QQ_BQ_C$ are $O_P$ and $O_Q$, respectively. Prove that $A,O_P,O_Q,R$ are concylic. [i]proposed by andychang[/i]

Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?

Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same? (A Shapovalov)

Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent. (N. Beluhov)

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

[b]p1.[/b] The entrance fee the county fair is $64$ cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay? [b]p2.[/b] At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position $A$ and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese? [b]p3.[/b] A clown stops you and poses a riddle. How many ways can you distribute $21$ identical balls into $3$ different boxes, with at least $4$ balls in the first box and at least $1$ ball in the second box? [b]p4.[/b] Watch out for the pig. How many sets $S$ of positive integers are there such that the product of all the elements of the set is $125970$? [b]p5.[/b] A good word is a word consisting of two letters $A$, $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length $8$. [b]p6.[/b] Evaluate $\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}$ without looking. [b]p7.[/b] There is nothing wrong with being odd. Of the first $2006$ Fibonacci numbers ($F_1 = 1$, $F_2 = 1$), how many of them are even? [b]p8.[/b] Let $f$ be a function satisfying $f (x) + 2f (27- x) = x$. Find $f (11)$. [b]p9.[/b] Let $A$, $B$, $C$ denote digits in decimal representation. Given that $A$ is prime and $A -B = 4$, nd $(A,B,C)$ such that $AAABBBC$ is a prime. [b]p10.[/b] Given $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ , find $\frac{x^8+y^8}{x^8-y^8}$ in term of $k$. [b]p11.[/b] Let $a_i \in \{-1, 0, 1\}$ for each $i = 1, 2, 3, ..., 2007$. Find the least possible value for $\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j$. [b]p12.[/b] Find all integer solutions $x$ to $x^2 + 615 = 2^n$ for any integer $n \ge 1$. [b]p13.[/b] Suppose a parabola $y = x^2 - ax - 1$ intersects the coordinate axes at three points $A$, $B$, and $C$. The circumcircle of the triangle $ABC$ intersects the $y$ - axis again at point $D = (0, t)$. Find the value of $t$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$. Let $M$ be the midpoint of side $BC$, and let $D'$ be the reflection of $D$ over $M$. Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$. Prove that $\angle MHG = 90^\circ$. [i]Proposed by Daniel Hu.[/i]

Three faces of a rectangular prism have diagonal lengths of $7$, $8$, and $9$ inches. How many cubic inches are in the volume of the rectangular prism? $\text{(A) }48\sqrt{11}\qquad\text{(B) }160\qquad\text{(C) }14\sqrt{95}\qquad\text{(D) }35\sqrt{15}\qquad\text{(E) }504$

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is $\textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad$

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]

In an $m\times n$ table there is a policeman in cell $(1,1)$, and there is a thief in cell $(i,j)$. A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which $(i,j)$ the policeman can catch the thief?

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: ${{ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15}\qquad\textbf{(E)}\ 18} $

The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle. - My rectangle ... - kid sobbed. - There is something strange about this figure! - Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of ​​the rectangle was, and the area is the same as was the perimeter! What size square did Carlson cut out?

A container is shaped like a right circular cone with base diameter $18$ and height $12$. The vertex of the container is pointing down, and the container is open at the top. Four spheres, each with radius $3$, are placed inside the container as shown. The first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are placed so they are each tangent to the cone and tangent to the rst sphere, and the second and fourth spheres are each tangent to the third sphere. The volume of the tetrahedron whose vertices are at the centers of the spheres is $K$. Find $K^2$. [img]https://cdn.artofproblemsolving.com/attachments/9/c/648ec2cf0f0c2f023cd00b1c0595a9396d0ddc.png[/img]

Acute triangle $ABC$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $BC.$ Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM.$ Segments $PQ$ and $BC$ intersect at $S$. Suppose that $BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}},$ and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^2$ can be expressed as $\tfrac ab$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

Points $D$ and $E$ are chosen on the sides $AB$ and $AC$ of the triangle $ABC$ in such a way that if $F$ is the intersection point of $BE$ and $CD$, then $AE + EF = AD + DF$. Prove that $AC + CF = AB + BF.$

Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular drawn from point $F$ on $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$. KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse

Given a segment $ AB $ and a line $ m $ parallel to this segment. Find the midpoint of the segment $ AB $ using only a ruler, i.e. drawing only straight lines.

Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo: Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let $F_A$ be the (associated with $\Omega$) Feuerbach point of the triangle formed by the tangents to $\Omega$ at $B,C,D$, that is, the point of tangency of $\Omega$ and the nine-point circle of that triangle. Define $F_B, F_C, F_D$ similarly. Let $A'$ be the intersection of the tangents to $\Omega$ at $A$ and $F_A$. Define $B', C', D'$ similarly. Prove that quadrilaterals $ABCD$ and $A'B'C'D'$ are similar