Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.

A circle $k$ with center $O$ and radius $r$ and a line $p$ which has no common points with $k$, are given. Let $E$ be the foot of the perpendicular from $O$ to $p$. Let $M$ be an arbitrary point on $p$, distinct from $E$. The tangents from the point $M$ to the circle $k$ are $MA$ and $MB$. If $H$ is the intersection of $AB$ and $OE$, then prove that $OH=\frac{r^2}{OE}$.

Let a cyclic quadrilateral $ ABCD$, $ AC \cap BD \equal{} E$ and let a circle $ \Gamma$ internally tangent to the arch $ BC$ (that not contain $ D$) in $ T$ and tangent to $ BE$ and $ CE$. Call $ R$ the point where the angle bisector of $ \angle ABC$ meet the angle bisector of $ \angle BCD$ and $ S$ the incenter of $ BCE$. Prove that $ R$, $ S$ and $ T$ are collinear. [i](Gabriel Giorgieri)[/i]

Let $ABCD$ be a parallelogram. Suppose that there exists a point $P$ in the interior of the parallelogram which is on the perpendicular bisector of $AB$ and such that $\angle PBA = \angle ADP$ Show that $\angle CPD = 2 \angle BAP$

Given a square sheet of paper with side $1$. Measure on this sheet a distance of $ 5/6$. (The sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).

Let $ ABC$ be a triangle. Prove that there is a unique point $ U$ in the plane of $ ABC$ such that there exist real numbers $ \alpha, \beta, \gamma, \delta$ not all zero, such that \[ \alpha PL^2 \plus{} \beta PM^2 \plus{} \gamma PN^2 \plus{} \delta UP^2\] is constant for all points $ P$ of the plane, where $ L,M,N$ are the feet of the perpendiculars from $ P$ to $ BC,CA,AB$ respectively. Identify $ U.$

The diagram below shows twelve $30-60-90$ triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.8)); pair point=(-sqrt(3),0); pair past,unit; path line; for(int i=0;i<=12;++i) { past = point; line=past--origin; unit=waypoint(line,1/200); point=extension(past,rotate(90,past)*unit,origin,dir(180-30*i)); if (i == 4) { filldraw(origin--past--point--cycle,gray(0.7)); } else if (i==12) { filldraw(origin--past--point--cycle,gray(0.7)); } else { draw(origin--past--point); } } draw(origin--point); [/asy]

A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations \[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\] Prove that $(ABC)$ is equilateral and $D$ is its center.

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.

In a trapezoid $ABCD$ with $AD$ parallel to $BC$ points $E, F$ are on sides $AB, CD$ respectively. $A_1, C_1$ are on $AD,BC$ such that $A_1, E, F, A$ lie on a circle and so do $C_1, E, F, C$. Prove that lines $A_1C_1, BD, EF$ are concurrent.

[b]3A.[/b] Sudoku, the popular math game that caught on internationally before making its way here to the United States, is a game of logic based on a grid of $9$ rows and $9$ columns. This grid is subdivided into $9$ squares (“subgrids”) of length $3$. A successfully completed Sudoku puzzle fills this grid with the numbers $1$ through $9$ such that each number appears only once in each row, column, and individual $3 \times 3$ subgrid. Each Sudoku puzzle has one and only one correct solution. Complete the following Sudoku puzzle, and find the sum of the numbers represented by $X, Y$, and $Z$ in the grid. [i](1 point)[/i] $\begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline & & 2 & 9 & 7 & 4 & & & \\ \hline & Z & & & & & & 5 & 7 \\ \hline & & & & & & Y & & \\ \hline & & 4 & & 5 & & & & 2 \\ \hline & & 9 & X & 1 & & 6 & & \\ \hline 8 & & & & 3 & & 4 & & \\ \hline & & & & & & & & \\ \hline 1 & 3 & & & & & & & \\ \hline & & & 6 & 8 & 2 & 9 & & \\ \hline \end{tabular}$ [b]3B.[/b] Let $A$ equal the correct answer from [b]3A[/b]. In triangle $WXY$, $tan \angle YWX= (A + 8) / .5A$, and the altitude from $W$ divides $XY$ into segments of $3$ and $A + 3$. What is the sum of the digits of the square of the area of the triangle? [i](2 points)[/i] [b]3C.[/b] Let $B$ equal the correct answer from [b]3B[/b]. If a student team taking the $2005$ iTest solves $B$ problems correctly, and the probability that this student team makes over a $18$ is $x/y$ where $x$ and $y$ are relatively prime, find $x + y$. Assume that each chain reaction question – all $3$ parts it contains – counts as a single problem. Also assume that the student team does not attempt any tiebreakers. [i](4 points)[/i] [i][Note for problem 3C beacuse you might not know how points are given at that iTest: Part A (aka Short Answer), has 40 problems of 1 point each, total 40 Part B (aka Chain Reaction), has 3 problems of 7,6,7 points each, total 20 Part C (aka Long Answer), has 5 problems of 8 point each, total 40 all 3 parts add to 100 points totally ([url=https://artofproblemsolving.com/community/c3176431_itest_2005]here [/url] is that test)][/i] [hide=ANSWER KEY]3A.14 3B. 4 3C. 6563 [/hide]

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

The perimeter of the triangle $ABC$ is equal to $2p$, the length of the side$ AC$ is equal to $b$, the angle $ABC$ is equal to $\beta$. A circle with center at point $O$, inscribed in this triangle, touches the side $BC$ at point $K$. Calculate the area of ​​the triangle $BOK$.

If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be $ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$ $ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$

3. Let $ABC$ be a triangle with $AB=30$, $BC=14$, and $CA=26$. Let $N$ be the center of the equilateral triangle constructed externally on side $AB$. Let $M$ be the center of the square constructed externally on side $BC$. Given that the area of quadrilateral $ACMN$ can be expressed as $a+b\sqrt{c}$ for positive integers $a$, $b$ and $c$ such that $c$ is not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by winnertakeover[/i]

A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.

In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$? [asy] real x=2sqrt(2); real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2)); pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = (4,0); C = (4,4); D = (0,4); E = (x,0); F = (4,y); G = (y,4); H = (0,x); I = F + z * dir(225); J = G + z * dir(225); draw(A--B--C--D--A); draw(H--E); draw(J--G^^F--I); draw(rightanglemark(G, J, I), linewidth(.5)); draw(rightanglemark(F, I, E), linewidth(.5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$E$", E, S); dot("$F$", F, dir(0)); dot("$G$", G, N); dot("$H$", H, W); dot("$I$", I, SW); dot("$J$", J, SW); [/asy] $\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2$

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Lines that are drawn perpendicular to the faces of a triangular pyramid through the centers of the inscribed circles intersect at one point. Prove that the sums of the opposite edges of such a pyramid are equal to each other.

A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures.

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

Lines from the vertices of a unit square are drawn to the midpoints of the sides as shown in the figure below. What is the area of quadrilateral $ABCD$? Express your answer in simplest terms. [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(1,0.5)); draw((1,0)--(0.5,1)); draw((1,1)--(0,0.5)); draw((0,1)--(0.5,0)); label("$A$",(0.2,0.6),N); label("$B$",(0.4,0.2),W); label("$C$",(0.8,0.4),S); label("$D$",(0.6,0.8),E); [/asy] $\text{(A) }\frac{\sqrt{2}}{9}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{\sqrt{3}}{9}\qquad\text{(D) }\frac{\sqrt{8}}{8}\qquad\text{(E) }\frac{1}{5}$

A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?