Temerant is a spherical planet with radius $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as a $\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

The diagonals of a convex quadrilateral $ABCD$ intersect in $E$. Let $S_1, S_2, S_3$ and $S_4$ the areas of the triangles $AEB, BEC, CED, DEA$ respectively. Prove that, if exists real numbers $w, x, y$ and $z$ such that $$S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,$$ then $E$ is the midpoint of $AC$ or $E$ is the midpoint of $BD$.

Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.

The points $S$ lie on side $AB$, $T$ on side $BC$, and $U$ on side $CA$ of a triangle so that the following applies: $\overline{AS} : \overline{SB} = 1 : 2$, $\overline{BT} : \overline{TC} = 2 : 3$ and $\overline{CU} : \overline{UA} = 3 : 1$. Construct the triangle $ABC$ if only the points $S, T$ and $U$ are given.

sorry if this has been posted before . given a fixed circle $(O)$ and two fixed point $B,C$ on it.point A varies on circle $(O)$. let $I$ be the midpoint of $BC$ and $H$ be the orthocenter of $\triangle ABC$. ray $IH$ meet $(O)$ at $K$ ,$AH$ meet $BC$ at $D$ ,$KD$ meet $(O)$ at $M$ .a line pass $M$ and perpendicular to $BC$ meet $AI$ at $N$. a) prove that $N$ varies on a fixed circle. b) a circle pass $N$ and tangent to $AK$ at $A$ cut $AB,AC$ at $P,Q$. let $J$ be the midpoint of $PQ$ .prove that $AJ$ pass through a fixed point.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Circles $\omega_1$ and $\omega_2$ meet at point $P,Q$. Let $O$ be the common point of external tangents of $\omega_1$ and $\omega_2$. A line passing through $O$ meets $\omega_1$ and $\omega_2$ at points $A,B$ located on the same side with respect to line segment $PQ$.The line $PA$ meets $\omega_2$ for the second time at $C$ and the line $QB$ meets $\omega_1$ for the second time at $D$. Prove that $O-C-D$ are collinear.

Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]

[u]Round 1[/u] [b]p1.[/b] Every angle of a regular polygon has degree measure $179.99$ degrees. How many sides does it have? [b]p2.[/b] What is $\frac{1}{20} + \frac{1}{1}+ \frac{1}{5}$ ? [b]p3.[/b] If the area bounded by the lines $y = 0$, $x = 0$, and $x = 3$ and the curve $y = f(x)$ is $10$ units, what is the area bounded by $y = 0$, $x = 0$, $x = 6$, and $y = f(x/2)$? [u]Round 2[/u] [b]p4.[/b] How many ways can $42$ be expressed as the sum of $2$ or more consecutive positive integers? [b]p5.[/b] How many integers less than or equal to $2015$ can be expressed as the sum of $2$ (not necessarily distinct) powers of two? [b]p6.[/b] $p,q$, and $q^2 - p^2$ are all prime. What is $pq$? [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 + ax + a$ be a polynomial with integer roots, where $a$ is an integer. What are all the possible values of $a$? [b]p8.[/b] In a given right triangle, the perimeter is $30$ and the sum of the squares of the sides is $338$. Find the lengths of the three sides. [b]p9.[/b] Each of the $6$ main diagonals of a regular hexagon is drawn, resulting in $6$ triangles. Each of those triangles is then split into $4$ equilateral triangles by connecting the midpoints of the $3$ sides. How many triangles are in the resulting figure? [u]Round 4[/u] [b]p10.[/b] Let $f = 5x+3y$, where $x$ and $y$ are positive real numbers such that $xy$ is $100$. Find the minimum possible value of $f$. [b]p11.[/b] An integer is called "Awesome" if its base $8$ expression contains the digit string $17$ at any point (i.e. if it ever has a $1$ followed immediately by a $7$). How many integers from $1$ to $500$ (base $10$) inclusive are Awesome? [b]p12.[/b] A certain pool table is a rectangle measuring $15 \times 24$ feet, with $4$ holes, one at each vertex. When playing pool, Joe decides that a ball has to hit at least $2$ sides before getting into a hole or else the shot does not count. What is the minimum distance a ball can travel after being hit on this table if it was hit at a vertex (assume it only stops after going into a hole) such that the shot counts? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}\plus{} \tfrac{1}{PC}$ is a maximum. [asy] size(200); defaultpen(linewidth(0.7)); pair A = origin, B = (5,0), C = (4.2,3), P = waypoint(B--C,0.65); pair Bp = 1.3 * B, Cp = 1.2 * C; draw(A--B--C--A^^Bp--A--Cp); dot(P); label("$A$",A,W); label("$B$",B,S); label("$C$",C,dir(B--C)); label("$P$",P,dir(A--P)); [/asy]

${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$, respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$. If we join the four midpoints of the sides, ${X}$, ${Y}$, ${Z}$, and ${T}$, to ${O}$, we form four quadrilaterals: ${OXBY}$, ${OYCZ}$, ${OZDT}$, and ${OTAX}$. Prove that the four newly formed quadrilaterals have the same areas.

The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is $\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}$

Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.

In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear. [i]Tiger Zhang[/i]

Let $ABC$ a triangle and let $\omega$ be its circumcircle. Let $E{}$ be the midpoint of the minor arc $BC$ of $\omega$, and $M{}$ the midpoint of $BC$. Let $V$ be the other point of intersection of $AM$ with $\omega$, $F{}$ the point of intersection of $AE$ with $BC$, $X{}$ the other point of intersection of the circumcircle of $FEM$ with $\omega$, $X'$ the reflection of $V{}$ with respect to $M{}$, $A'{}$ the foot of the perpendicular from $A{}$ to $BC$ and $S{}$ the other point of intersection of $XA'$ with $\omega$. If $Z \in \omega$ with $Z\neq X$ is such that $AX = AZ$, then prove that $S, X'$ and $Z{}$ are collinear.

Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.

Using a ruler with a length of $20$ cm and a compass with a maximum deviation of $10$ cm to connect the segment given two points lying at a distance of $1$ m.

Let $\vartriangle ABC$ be a triangle with $AB = 15$, $AC = 13$, $BC = 14$, and circumcenter $O$. Let $\ell$ be the line through $A$ perpendicular to segment $BC$. Let the circumcircle of $\vartriangle AOB$ and the circumcircle of $\vartriangle AOC$ intersect $\ell$ at points $X$ and $Y$ (other than $A$), respectively. Compute the length of $\overline{XY}$ .

Let $ ABC$ be an acute triangle, and let $ D$ and $ E$ be the feet of the altitudes drawn from vertexes $ A$ and $ B$, respectively. Show that if, \[ Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]\] then, the triangle is isosceles.

Let \( ABC \) be a triangle and \( D \) the foot of the altitude from \( A \). Let \( M \) be a point such that \( MB = MC \). Let \( E \) and \( F \) be the intersections of the circumcircle of \( BMD \) and \( CMD \) with \( AD \). Let \( G \) and \( H \) be the intersections of \( MB \) and \( MC \) with \( AD \). Prove that \( EG = FH \).

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece? $\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$

Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear. [i]Rudi Adha Prihandoko, Bandung[/i]