Convex polyhedron $M$ with triangular faces is cut into tetrahedrons; all the vertices of the tetrahedrons are the vertices of the polyhedron, and any two tetrahedrons either do not intersect, or they intersect along a common vertex, common edge, or common face. Prove that it it's not possible that each tetrahedron has exactly one face on the surface of $M$.

Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$. (a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$. (b) Determine the length of the perimeter of this hexagon when $a + b > c$.

The square $ABCD$ is inscribed in a circle with center $O$. Let $E$ be the midpoint of $AD$. The line $CE$ meets the circle again at $F$. The lines $FB$ and $AD$ meet at $H$. Prove $HD = 2AH$

Define a $\emph{big circle}$ in a sphere as a circle that has two diametrically oposite points of the sphere in it. Suppose $(AB)$ as the big circle that passes through $A$ and $B$. Also, let a $\emph{Spheric Triangle}$ be $3$ connected by big circles. The angle between two circles that intersect is defined by the angle between the two tangent lines from the intersection point through the two circles in their respective planes. Define also $\angle XYZ$ the angle between $(XY)$ and $(YZ)$. Two circles are tangent if the angle between them is 0. All the points in the following problem are in a sphere S. Let $\Delta ABC$ be a spheric triangle with all its angles $<90^{\circ}$ such that there is a circle $\omega$ tangent to $(BC)$,$(CA)$,$(AB)$ in $D,E,F$. Show that there is $P\in S$ with $\angle PAB=\angle DAC$, $\angle PCA=\angle FCB$, $\angle PBA=\angle EBC$.

$ABC$ is a right triangle with $AC=3$, $BC=4$, $AB=5$. Squares are erected externally on the sides of the triangle. Evaluate the area of hexagon $PQRSTU$.

Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

$E$ is a point on the diagonal $BD$ of the square $ABCD$. Show that the points $A, E$ and the circumcenters of $ABE$ and $ADE$ form a square.

In triangle $ABC$, $AB \ne AC$. Let $AF$, $AP$ and $AT$ be the median, angle bisector and altitude from vertex $A$, with $F, P$ and $T$ on $BG$ or its extension. (a) Prove that $P$ always lies between$ F$ and $T$. (b) Prove that $\angle FAP < \angle PAT$ if $ABC$ is an acute triangle.

Let us call a pentagon curved, if all its sides are arcs of some circles. Are there exist a curved pentagon $P$ and a point $A$ on its boundary so that any straight line passing through $A$ divides perimeter of $P$ into two parts of the same length? [i](7 points)[/i]

Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that \[ \frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} . \] [i]Bogdan Enescu[/i]

Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None of above} $

Square $ AIME$ has sides of length $ 10$ units. Isosceles triangle $ GEM$ has base $ EM$, and the area common to triangle $ GEM$ and square $ AIME$ is $ 80$ square units. Find the length of the altitude to $ EM$ in $ \triangle GEM$.

Let $P$ an interior point of an equilateral triangle $ABC$. Prove that there exists triangle with sides $PA , PB , PC$ . Babis

Let $ L $ be the set of all polylines $ ABCDA $, where $ A, B, C, D $ are different vertices of a fixed regular $1985$ -gon. We randomly select a polyline from the set $L$. Calculate the probability that it is the side of a convex quadrilateral.

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

Let \(ABC\) be a right-angled triangle with \(\angle B=90^{\circ}\). Let \(I\) be the incentre if \(ABC\). Extend \(AI\) and \(CI\); let them intersect \(BC\) in \(D\) and \(AB\) in \(E\) respectively. Draw a line perpendicular to \(AI\) at \(I\) to meet \(AC\) in \(J\), draw a line perpendicular to \(CI\) at \(I\) to meet \(AC\) at \(K\). Suppose \(DJ=EK\). Prove that \(BA=BC\).

In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$. [i]Ray Li[/i]

Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$

From which two statements about the trapezoid follows the third: 1) the trapezoid is tangential, 2) the trapezoid is right, 3) its area is equal to the product of the bases?

Points $A' ,B' ,C'$ on the respective sides $BC,CA,AB$ of triangle $ABC$ satisfy $\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k$. The lines $AA' ,BB' ,CC' $ form a triangle $A_1B_1C_1$ (possibly degenerate). Given $k$ and the area $S$ of $\triangle ABC$, compute the area of $\triangle A_1B_1C_1$.

[u]Set 6[/u] [b]B16.[/b] Let $\ell_r$ denote the line $x + ry + r^2 = 420$. Jeffrey draws the lines $\ell_a$ and $\ell_b$ and calculates their single intersection point. [b]B17.[/b] Let set $L$ consist of lines of the form $3x + 2ay = 60a + 48$ across all real constants a. For every line $\ell$ in $L$, the point on $\ell$ closest to the origin is in set $T$ . The area enclosed by the locus of all the points in $T$ can be expressed in the form nπ for some positive integer $n$. Compute $n$. [b]B18.[/b] What is remainder when the $2020$-digit number $202020 ... 20$ is divided by $275$? [u]Set 7[/u] [b]B19.[/b] Consider right triangle $\vartriangle ABC$ where $\angle ABC = 90^o$, $\angle ACB = 30^o$, and $AC = 10$. Suppose a beam of light is shot out from point $A$. It bounces off side $BC$ and then bounces off side $AC$, and then hits point $B$ and stops moving. If the beam of light travelled a distance of $d$, then compute $d^2$. [b]B20.[/b] Let $S$ be the set of all three digit numbers whose digits sum to $12$. What is the sum of all the elements in $S$? [b]B21.[/b] Consider all ordered pairs $(m, n)$ where $m$ is a positive integer and $n$ is an integer that satisfy $$m! = 3n^2 + 6n + 15,$$ where $m! = m \times (m - 1) \times ... \times 1$. Determine the product of all possible values of $n$. [u]Set 8[/u] [b]B22.[/b] Compute the number of ordered pairs of integers $(m, n)$ satisfying $1000 > m > n > 0$ and $6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)$. [b]B23.[/b] Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? [b]B24.[/b] Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$? [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]B25.[/b] Submit one of the following ten numbers: $$3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.$$ The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number $27$, you would receive $\left\lceil \frac{27}{5}\right\rceil = 6$ points. [b]B26.[/b] Submit any integer from $1$ to $1,000,000$, inclusive. The standard deviation $\sigma$ of all responses $x_i$ to this question is computed by first taking the arithmetic mean $\mu$ of all responses, then taking the square root of average of $(x_i -\mu)^2$ over all $i$. More, precisely, if there are $N$ responses, then $$\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.$$ For this problem, your goal is to estimate the standard deviation of all responses. An estimate of $e$ gives $\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}$ points. [b]B27.[/b] For a positive integer $n$, let $f(n)$ denote the number of distinct nonzero exponents in the prime factorization of $n$. For example, $f(36) = f(2^2 \times 3^2) = 1$ and $f(72) = f(2^3 \times 3^2) = 2$. Estimate $N = f(2) + f(3) +.. + f(10000)$. An estimate of $e$ gives $\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}$ points. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a square $ABCD$ with side $1$, and a square inside $ABCD$ with side $x$, find (in terms of $x$) the radio $r$ of the circle tangent to two sides of $ABCD$ and touches the square with side $x$. (See picture).

Let $N$ be a point inside the triangle $ABC$. Through the midpoints of the segments $AN, BN$, and $CN$ the lines parallel to the opposite sides of $\triangle ABC$ are constructed. Let $AN, BN$, and $CN$ be the intersection points of these lines. If $N$ is the orthocenter of the triangle $ABC$, prove that the nine-point circles of $\triangle ABC$ and $\triangle A_NB_NC_N$ coincide. [hide="Remark."]Remark. The statement of the original problem was that the nine-point circles of the triangles $A_NB_NC_N$ and $A_MB_MC_M$ coincide, where $N$ and $M$ are the orthocenter and the centroid of $ABC$. This statement is false.[/hide]

Let be a triangle $ ABC $ that has an angle of $ 120^{\circ } . $ Bisectors of all three angles meet the sides of the triangle at $ A',B',C'. $ Prove $ A'B'C' $ is a right triangle.

Let $ABC$ be an acute triangle with $AB<AC$. The angle bisector of $BAC$ intersects the side $BC$ and the circumcircle of $ABC$ at $D$ and $M\neq A$, respectively. Points $X$ and $Y$ are chosen so that $MX \perp AB$, $BX \perp MB$, $MY \perp AC$, and $CY \perp MC$. Prove that the points $X,D,Y$ are collinear.