Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves

In the scalene acute triangle $ABC$, $O$ is the circumcenter. $AD, BE, CF$ are three altitudes. And $H$ is the orthocenter. Let $G$ be the reflection point of $O$ through $BC$. Draw the diameter $EK$ in $\odot (GHE)$, and the diameter $FL$ in $\odot (GHF)$. a) If $AK, AL$ and $DE, DF$ intersect at $U, V$ respectively, prove that $UV\parallel EF$. b) Suppose $S$ is the intersection of the two tangents of the circumscribed circle of $\triangle ABC$ at $B$ and $C$. $T$ is the intersection of $DS$ and $HG$. And $M,N$ are the projection of $H$ on $TE,TF$ respectively. Prove that $M,N,E,F$ are concyclic.

[b]p1.[/b] What is $(2 + 4 + ... + 20) - (1 + 3 + ...+ 19)$? [b]p2.[/b] Two ants start on opposite vertices of a dodecagon ($12$-gon). Each second, they randomly move to an adjacent vertex. What is the probability they meet after four moves? [b]p3.[/b] How many distinct $8$-letter strings can be made using $8$ of the $9$ letters from the words $FORK$ and $KNIFE$ (e.g., $FORKNIFE$)? [b]p4.[/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 - BE^2$? [b][color=#f00](mistyped p4)[/color][/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 + BE^2$? [b]p5.[/b] You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter. [b]p6.[/b] Let $\omega_1$ and $\omega_2$ be two circles with centers $A$ and $B$ and radii $3$ and $13$, respectively. Suppose $AB = 10$ and that $C$ is the midpoint of $AB$. Let $\ell$ be a line that passes through $C$ and is tangent to $\omega_1$ at $P$. Given that $\ell$ intersects $\omega_2$ at $X$ and $Y$ such that $XP < Y P$, what is $XP$? [b]p7.[/b] Let $f(x)$ be a cubic polynomial. Given that $f(1) = 13$, $f(4) = 19$, $f(7) = 7$, and $f(10) = 13$, find $f(13)$. [b]p8.[/b] For all integers $0 \le n \le 202$ not divisible by seven, define $f(n) = \{\sqrt{7n}\}$. For what value $n$ does $f(n)$ take its minimum value? (Note: $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) [b]p9.[/b] Let $ABC$ be a triangle with $AB = 14$ and $AC = 25$. Let the incenter of $ABC$ be $I$. Let line $AI$ intersect the circumcircle of $BIC$ at $D$ (different from $I$). Given that line $DC$ is tangent to the circumcircle of $ABC$, find the area of triangle $BCD$. [b]p10.[/b] Evaluate the infinite sum $$\frac{4^2 + 3}{1 \cdot 3 \cdot 5 \cdot 7} +\frac{6^2 + 3}{3 \cdot 5 \cdot 7 \cdot 9}+\frac{8^2 + 3}{5 \cdot 7 \cdot 9 \cdot 11}+ ...$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from $A$ on$ CM$ with $CM$ and $BC$ respectively. Prove that a) $\angle AQC = \angle PQB$ b) $\angle DRQ = 45^o$

Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Let $ABC$ be a given acute triangle, in which $BC$ is the longest side. Let $H$ be the orthocenter of the triangle, and let $D$ and $E$ be the feet of the altitudes from $B$ and $C$, respectively. Let $F$ and $G$ be the midpoints of sides $AB$ and $AC$, respectively. $X$ is the point of intersection of lines $DF$ and $EG$. Let $O_1$ and $O_2$ be the circumcenters of triangles $EFX$ and $DGX$, respectively. Finally, $M$ is the midpoint of line segment $O_1O_2$. Prove that points $X, H$ and $M$ are collinear.

A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

Which pattern of identical squares could NOT be folded along the lines shown to form a cube? [asy] unitsize(12); draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle); draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)); draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle); draw((7,-1)--(8,-1)--(8,-2)--(7,-2)); draw((9,-1)--(9,-2)); draw((10,-1)--(10,-2)); draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle); draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1)); draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle); draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3)); draw((24,-2)--(24,-3)); draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle); draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1)); draw((29,-1)--(29,-2)); label("(A)",(0,-0.5),W); label("(B)",(7,-0.5),W); label("(C)",(14,-0.5),W); label("(D)",(21,-0.5),W); label("(E)",(28,-0.5),W); [/asy]

In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric. [i]Proposed by: Géza Kós, Budapest[/i]

A circle of radius $4$ is inscribed in a triangle $ABC$. We call $D$ the touchpoint between the circle and side BC. Let $CD =8$, $DB= 10$. What is the length of the sides $AB$ and $AC$?

The vertices of a triangle with sides $a\ge b\ge c$ are centers of three circles, such that no two of the circles have common interior points and none contains any other vertex of the triangle. Determine the maximum possible total area of these three circles.

Let $ ABC$ be a triangle with $ BC \equal{} 2007$, $ CA \equal{} 2008$, $ AB \equal{} 2009$. Let $ \omega$ be an excircle of $ ABC$ that touches the line segment $ BC$ at $ D$, and touches extensions of lines $ AC$ and $ AB$ at $ E$ and $ F$, respectively (so that $ C$ lies on segment $ AE$ and $ B$ lies on segment $ AF$). Let $ O$ be the center of $ \omega$. Let $ \ell$ be the line through $ O$ perpendicular to $ AD$. Let $ \ell$ meet line $ EF$ at $ G$. Compute the length $ DG$.

An acute-angled triangle $ABC$ has altitudes $AD, BE$ and $CF$. Let $Q$ be an interior point of the segment $AD$, and let the circumcircles of the triangles $QDF$ and $QDE$ meet the line $BC$ again at points $X$ and $Y$ , respectively. Prove that $BX = CY$ .

Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$. The resulting three smaller triangles have areas $1$, $4$, and $9$. Find the area of $\triangle ABC$. [asy] defaultpen(linewidth(0.7)); size(120); pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); } pair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2); draw(A--B--C--cycle); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7)); filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7)); filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]

Find the area of the equilateral triangle that includes vertices at $\left(-3,5\right)$ and $\left(-5,9\right)$. $\text{(A) }3\sqrt{3}\qquad\text{(B) }10\sqrt{3}\qquad\text{(C) }\sqrt{30}\qquad\text{(D) }2\sqrt{15}\qquad\text{(E) }5\sqrt{3}$

A rhombus is formed by two radii and two chords of a circle whose radius is $ 16$ feet. The area of the rhombus in square feet is: $ \textbf{(A)}\ 128 \qquad\textbf{(B)}\ 128\sqrt {3} \qquad\textbf{(C)}\ 256 \qquad\textbf{(D)}\ 512 \qquad\textbf{(E)}\ 512\sqrt {3}$

Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$

Which is the least positive integer $n$ for which it is possible to find a (non-degenerate) $n$-gon with sidelengths $1, 2,. . . , n$, and where all vertices have integer coordinates?

In a triangle $ABC$, points $P, Q$ and $ R$ distinct from the vertices of the triangle are chosen on sides $AB, BC$ and $CA$, respectively. The circumcircles of the triangles $APR$, $BPQ$, and $CQR$ are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle $ABC$.

A circle can be drawn around the quadrilateral $ABCD$. $K$ is a point on the diagonal $BD$ . The straight line $CK$ intersects the side $AD$ at the point $M$. Prove that the circles circumscribed around the triangles $BCK$ and $ACM$ are tangent.

[b]3.[/b] Let $k$ and $K$ be concentric circles on the plane, and let $k$ be contained inside $K$. Assume that $k$ is covered by a finite system of convex angular domains with vertices on $K$. Prove that the sum of the angles of the domains is not less than the angle under which $k$ can be seen from a point of $K$. ([b]G.38[/b]) [Zs.. Páles]

Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus $ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.

Let $A$ be point in the exterior of the circle $\mathcal C$. Two lines passing through $A$ intersect the circle $\mathcal C$ in points $B$ and $C$ (with $B$ between $A$ and $C$) respectively in $D$ and $E$ (with $D$ between $A$ and $E$). The parallel from $D$ to $BC$ intersects the second time the circle $\mathcal C$ in $F$. Let $G$ be the second point of intersection between the circle $\mathcal C$ and the line $AF$ and $M$ the point in which the lines $AB$ and $EG$ intersect. Prove that \[ \frac 1{AM} = \frac 1{AB} + \frac 1{AC}. \]