Let $ABC$ be an acute triangle and let $X$ and $Y$ be points on the segments $AB$ and $AC$ such that $BX = CY$. If $I_{B}$ and $I_{C}$ are centers of inscribed circles in triangles $ABY$ and $ACX$, and $T$ is the second intersection point of the circumcircles of $ABY$ and $ACX$, show that: $$\frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}.$$ [i]Proposed by Nikola Velov[/i]

Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.

Let $ABCD$ be a unit square and let $P$ and $Q$ be points in the plane such that $Q$ is the circumcentre of triangle $BPC$ and $D$ be the circumcentre of triangle $PQA$. Find all possible values of the length of segment $PQ$.

The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

Circle I is circumscribed about a given square and circle II is inscribed in the given square. If $r$ is the ratio of the area of circle $I$ to that of circle $II$, then $r$ equals: $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt 3 \qquad \text{(D)} \ 2\sqrt 2 \qquad \text{(E)} \ 2\sqrt 3$

Given is the non-equilateral triangle $A_1A_2A_3$. $B_{ij}$ is the symmetric of $A_i$ wrt the inner bisector of $\angle A_j$. Prove that lines $B_{12}B_{21}$, $B_{13}B_{31}$ and $B_{23}B_{32}$ are parallel.

A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$.

[u]Round 6[/u] 16. If you roll four fair $6$-sided dice, what is the probability that at least three of them will show the same value. 17. In a tetrahedron with volume $1$, four congruent speres are placed each tangent to three walls and three other spheres. What is the radii of each of the spheres. 18. let $f(x)$ be twice the number of letters in $x$. What is the sum of the unique $x,y$ such that $x \ne y$ and $f(x)=y$ and $f(y)=x$. [u]Round 7[/u] [b]p19.[/b] How many $4$ digit numbers with distinct digits $ABCD$ with $A$ not equal to $0$ are divisible by $11$? [b]p20.[/b] How many ($2$-dimensional) faces does a $2014$-dimensional hypercube have? [b]p21.[/b] How many subsets of the numbers $1,2,3,4...2^{2014}$ have a sum of $2014$ mod $2^{2014}$? [u]Round 8[/u] [b]p22.[/b] Two diagonals of a dodecagon measure $1$ unit and $2$ units. What is the area of this dodecagon? [b]p23.[/b] Square $ABCD$ has point $X$ on AB and $Y$ on $BC$ such that angle $ADX = 15$ degrees and angle $CDY = 30$ degrees. what is the degree measure of angle $DXY$? [b]p24.[/b] A $4\times 4$ grid has the numbers $1$ through $16$ placed in it, $1$ per cell, such that no adjacent boxes have cells adding to a number divisible by $3$. In how many ways is this possible? [u]Round 9[/u] [b]p25.[/b] Let $B$ and $C$ be the answers to $26$ and $27$ respectively.If $S(x)$ is the sum of the digits in $x$, what is the unique integer $A$ such that $S(A), S(B), S(C) \subset A,B,C$. [b]p26.[/b] Let $A$ and $C$ be the answers to $25$ and $27$ respectively. What is the third angle of a triangle with two of its angles equal to $A$ and $C$ degrees. [b]p27.[/b] Let $A$ and $B$ be the answers to $25$ and $26$ respectively. How many ways are there to put $A$ people in a line, with exactly $B$ places where a girl and a boy are next to each other. [u]Round 10[/u] [b]p28.[/b] What is the sum of all the squares of the digits to answers to problems on the individual, team, and theme rounds of this years LMT? If the correct answer is $N$ and you submit $M$, you will recieve $\lfloor 15 - 10  \times \log (M - N) \rfloor $. [b]p29.[/b] How many primes have all distinct digits, like $2$ or $109$ for example, but not $101$. If the correct answer is $N$ and you submit $M$, you will recieve $\left\lfloor 15 \min \left( \frac{M}{N} , \frac{N}{M} \right)\right\rfloor $. [b]p30.[/b] For this problem, you can use any $10$ mathematical symbols that you want, to try to achieve the highest possible finite number. (So "Twenty-one", " $\frac{12}{100} +843$" and "$\sum^{10}_{i=0} i^2 +1$" are all valid submissions.) If your team has the $N$th highest number, you will recieve $\max (16 - N, 0)$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let \( ABCDE \) be a convex pentagon whose vertices lie on a circle \( \Gamma \). The tangents to \( \Gamma \) at \( C \) and \( E \) intersect at \( X \), and the segments \( CE \) and \( AD \) intersect at \( Y \). Given that \( CE \) is perpendicular to \( BD \), that \( XY \) is parallel to \( BD \), that \( AY = BD \), and that \( \angle BAD = 30^\circ \), what is the measure of the angle \( \angle BDA \)? Proposed by João Gilberti Alves Tavares

Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

An equilateral triangle whose side is $ 2$ is divided into a triangle and a trapezoid by a line drawn parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original triangle, the length of the median of the trapezoid is: $ \textbf{(A)}\ \frac{\sqrt{6}}{2} \qquad \textbf{(B)}\ \sqrt{2} \qquad \textbf{(C)}\ 2\plus{}\sqrt{2} \qquad \textbf{(D)}\ \frac{2\plus{}\sqrt{2}}{2} \qquad \textbf{(E)}\ \frac{2\sqrt{3}\minus{}\sqrt{6}}{2}$

On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals. [i]A. Tereshin[/i]

$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively. Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$

Let $ABCD$ and $AEF G$ be unit squares such that the area of their intersection is $\frac{20}{21}$ . Given that $\angle BAE < 45^o$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

In quadrilateral $ABCD$, a circle $\omega$ is inscribed. A point $K$ is chosen on diagonal $AC$. Segment $BK$ intersects $\omega$ at a unique point $X$, and segment $DK$ intersects $\omega$ at a unique point $Y$. It turns out that $XY$ is the diameter of $\omega$. Prove that it is perpendicular to $AC$. [i]Proposed by Tseren Frantsuzov[/i]

A regular tetrahedron with $5$-inch edges weighs $2.5$ pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has $6$-inch edges? Express your answer as a decimal rounded to the nearest hundredth.

Let $ABC$ be a non-degenerate triangle in the euclidean plane. Define a sequence $(C_n)_{n=0}^\infty$ of points as follows: $C_0:=C$, and $C_{n+1}$ is the incenter of the triangle $ABC_n$. Find $\lim_{n\to\infty}C_n$.

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.

Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? [i]Proposed by Vlad Spătaru[/i]

Is it possible to divide the plane into polygons so that each polygon is transformed into itself under some rotation by $360/7$ degrees about some point? All sides of these polygons must be greater than $1$ cm. (A polygon is the part of a plane bounded by one non-self-intersect-ing closed broken line, not necessarily convex.) (A. Andjans, Riga)

Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$

Triangle $ABC,$ with $BC = 48,$ is inscribed in a circle $\Omega$ of radius $49\sqrt{3}.$ There is a unique circle $\omega$ that is tangent to $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Omega.$ Let $D,$ $E,$ and $F$ be the points at which $\omega$ is tangent to $\Omega,$ $\overline{AB},$ and $\overline{AC},$ respectively. The rays $\overrightarrow{DE}$ and $\overrightarrow{DF}$ intersect $\Omega$ at points $X$ and $Y,$ respectively, such that $X \neq D$ and $Y \neq D.$ Compute $XY.$

In the quadrilateral $ABCD$ the angles $B{}$ and $D{}$ are straight. The lines $AB{}$ and $DC{}$ intersect at $E$ and the lines $AD$ and $BC$ intersect at $F{}.$ The line passing through $B{}$ parallel to $C{}$D intersects the circumscribed circle $\omega$ of $ABF{}$ at $K{}$ and the segment $KE{}$ intersects $\omega$ at $P{}.$ Prove that the line $AP$ divides the segment $CE$ in half.

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.