Triangle $ABC$ has sidelengths $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with sidelengths $AG$, $BG$, and $CG$

Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?

Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, $BC = 6$. The angle bisector of $C$ intersects side $AB$ at $X$. Points $M$ and $N$ are drawn on sides $BC$ and $AC$, respectively, such that $\overline{XM} \parallel \overline{AC}$ and $\overline{XN} \parallel \overline{BC}$. Compute the length $MN$.

The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces are S1, S2, S3, S4, and its volume is V . Prove that 2 [S1 S2 S3 S4](1/2) > 3V [abcdef](1/6) this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf I was just wondering if someone could write it in LATEX form. [color=red]_____________________________________ EDIT by moderator: If you type[/color] [code]The edge lengths of a tetrahedron are $a, b, c, d, e, f,$ the areas of its faces are $S_1, S_2, S_3, S_4,$ and its volume is $V.$ Prove that $2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$[/code] [color=red]it shows up as:[/color] The edge lengths of a tetrahedron are $ a, b, c, d, e, f,$ the areas of its faces are $ S_1, S_2, S_3, S_4,$ and its volume is $ V.$ Prove that $ 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.

A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.

In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$. [asy] unitsize(1 cm); pair O; pair[] A, B; O = (0,0); A[1] = (0.5,-3); A[2] = (2,0); A[3] = (-0.2,0.5); A[4] = (-1,0); B[1] = A[2] - A[1]; B[2] = A[3] - A[2]; B[3] = A[4] - A[3]; B[4] = A[1] - A[4]; draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[2]--B[3]--B[4]--cycle); draw(O--B[1]); draw(O--B[2]); draw(O--B[3]); draw(O--B[4]); label("$A_1$", A[1], S); label("$A_2$", A[2], E); label("$A_3$", A[3], N); label("$A_4$", A[4], W); label("$B_1$", B[1], NE); label("$B_2$", B[2], W); label("$B_3$", B[3], SW); label("$B_4$", B[4], S); label("$O$", O, E); [/asy]

A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$

In a convex quadrilateral it is known $ABCD$ that $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}$ and $AD = BC$. Prove that from the lengths $DB$, $CA$ and $DC$, you can make a right triangle.

The midpoint $M$ of $[AC]$ of a triangle $\triangle ABC$ is between $C$ and the feet $H$ of the altitude from $B$. If $m(\widehat{ABH}) = m(\widehat{MBC})$, $m(\widehat{ACB}) = 15^{\circ}$, and $|HM|=2\sqrt{3}$, then $|AC|=?$ $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 5 \sqrt 2 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ \frac{16}{\sqrt3} \qquad \textbf{(E)}\ 10$

In a square of sidelength $7$, $51$ points are given. Show that there's a disk of radius $1$ covering at least $3$ of these points.

Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$. Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.

Let $h_a,h_b,h_c$ be the altitudes, and let $m_a,m_b,m_c$ be the medians of the acute triangle drawn to the sides $a, b, c$ respectively. Let $r$ and $R$ be the radii of the inscribed and circumscribed circles. Prove that $$\frac{m_a}{h_a}+\frac{m_b}{h_b}+\frac{m_c}{h_c} <1+\frac{R}{r}.$$

On a spherical surface there is a set $M{}$ with $n{}$ points with the property: for every point $A{}$ from $M{}$ there exist points $B$ and $C$ from $M{}$ such that the triangle $ABC$ is equilateral. For every equilateral triangle with vertexes in $M{}$ the perpendicular on its plane that goes through the geometric center of the other points from $M{}$. Prove that all these perpendiculars are concurrent.

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled. [i]Proposed by Alexander A. Polyansky, Russia[/i]

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

Given three squares as in the figure (where the vertex of B is touching square A --- the diagram had an error), where the largest square has area 1, and the area $ A$ is known. What is the area $ B$ of the smallest square? [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number4.jpg[/img] A. $ A/8$ B. $ \frac {A^2}{2}$ C. $ \frac {A^4}{4}$ D. $ A(1 \minus{} A)$ E. $ \frac {(1 \minus{} A)^2}{4}$

Suppose $ABCD$ is a convex quadrilateral satisfying $AB=BC$, $AC=BD$, $\angle ABD = 80^\circ$, and $\angle CBD = 20^\circ$. What is $\angle BCD$ in degrees?

The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1,B_1,C_1,D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [b]C1.[/b] Mr. Pham flips $2018$ coins. What is the difference between the maximum and minimum number of heads that can appear? [b]C2 / G1.[/b] Brandon wants to maximize $\frac{\Box}{\Box} +\Box$ by placing the numbers $1$, $2$, and $3$ in the boxes. If each number may only be used once, what is the maximum value attainable? [b]C3.[/b] Guang has $10$ cents consisting of pennies, nickels, and dimes. What are all the possible numbers of pennies he could have? [b]C4.[/b] The ninth edition of Campbell Biology has $1464$ pages. If Chris reads from the beginning of page $426$ to the end of page$449$, what fraction of the book has he read? [b]C5 / G2.[/b] The planet Vriky is a sphere with radius $50$ meters. Kyerk starts at the North Pole, walks straight along the surface of the sphere towards the equator, runs one full circle around the equator, and returns to the North Pole. How many meters did Kyerk travel in total throughout his journey? [b]C6 / G3.[/b] Mr. Pham is lazy and decides Stan’s quarter grade by randomly choosing an integer from $0$ to $100$ inclusive. However, according to school policy, if the quarter grade is less than or equal to $50$, then it is bumped up to $50$. What is the probability that Stan’s final quarter grade is $50$? [b]C7 / G5.[/b] What is the maximum (finite) number of points of intersection between the boundaries of a equilateral triangle of side length $1$ and a square of side length $20$? [b]C8.[/b] You enter the MBMT lottery, where contestants select three different integers from $1$ to $5$ (inclusive). The lottery randomly selects two winning numbers, and tickets that contain both of the winning numbers win. What is the probability that your ticket will win? [b]C9 / G7.[/b] Find a possible solution $(B, E, T)$ to the equation $THE + MBMT = 2018$, where $T, H, E, M, B$ represent distinct digits from $0$ to $9$. [b]C10.[/b] $ABCD$ is a unit square. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $AD$. $DE$ and $CF$ meet at $G$. Find the area of $\vartriangle EFG$. [b]C11.[/b] The eight numbers $2015$, $2016$, $2017$, $2018$, $2019$, $2020$, $2021$, and $2022$ are split into four groups of two such that the two numbers in each pair differ by a power of $2$. In how many different ways can this be done? [b]C12 / G4.[/b] We define a function f such that for all integers $n, k, x$, we have that $$f(n, kx) = k^n f(n, x) and f(n + 1, x) = xf(n, x).$$ If $f(1, k) = 2k$ for all integers $k$, then what is $f(3, 7)$? [b]C13 / G8.[/b] A sequence of positive integers is constructed such that each term is greater than the previous term, no term is a multiple of another term, and no digit is repeated in the entire sequence. An example of such a sequence would be $4$, $79$, $1035$. How long is the longest possible sequence that satisfies these rules? [b]C14 / G11.[/b] $ABC$ is an equilateral triangle of side length $8$. $P$ is a point on side AB. If $AC +CP = 5 \cdot AP$, find $AP$. [b]C15.[/b] What is the value of $(1) + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + 49 + 50)$? [b]G6.[/b] An ant is on a coordinate plane. It starts at $(0, 0)$ and takes one step each second in the North, South, East, or West direction. After $5$ steps, what is the probability that the ant is at the point $(2, 1)$? [b]G10.[/b] Find the set of real numbers $S$ so that $$\prod_{c\in S}(x^2 + cxy + y^2) = (x^2 - y^2)(x^{12} - y^{12}).$$ [b]G12.[/b] Given a function $f(x)$ such that $f(a + b) = f(a) + f(b) + 2ab$ and $f(3) = 0$, find $f\left( \frac12 \right)$. [b]G13.[/b] Badville is a city on the infinite Cartesian plane. It has $24$ roads emanating from the origin, with an angle of $15$ degrees between each road. It also has beltways, which are circles centered at the origin with any integer radius. There are no other roads in Badville. Steven wants to get from $(10, 0)$ to $(3, 3)$. What is the minimum distance he can take, only going on roads? [b]G14.[/b] Team $A$ and Team $B$ are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a $50\%$ chance of scoring $1$ point. Regardless of whether or not they score, the ball is given to the other team after they attempt to score. What is the probability that Team $A$ will score $5$ points before Team $B$ scores any? [b]G15.[/b] The twelve-digit integer $$\overline{A58B3602C91D},$$ where $A, B, C, D$ are digits with $A > 0$, is divisible by $10101$. Find $\overline{ABCD}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$. Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$). Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.

Adam is walking in the city. In order to get around a large building, he walks $12$ miles east and then $5$ miles north, then stop. His friend Neutrino, who can go through buildings, starts in the same place as Adam but walks in a straight line to where Adam stops. How much farther than Neutrino does Adam walk? $$ \mathrm a. ~ 1~\mathrm{mile}\qquad \mathrm b.~2 ~\mathrm{miles}\qquad \mathrm c. ~3~\mathrm{miles} \qquad \mathrm d. ~4~\mathrm{miles} \qquad \mathrm e. ~5~\mathrm{miles} $$

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. At point $A$ to $\omega_1$ and $\omega_2$ the tangents $\ell_1$ and $\ell_2$ are drawn respectively. The points $T_1$ and $T_2$ are chosen respectively on the circles $\omega_1$ and $\omega_2$ so that the angular measures of the arcs $T_1A$ and $AT_2$ are equal (the measure of the circular arc is calculated clockwise). The tangent $t_1$ at the point $ T_1$ to the circle $\omega_1$ intersects $\ell_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle $\omega_2$ intersects $\ell_1$ at point $M_2$. Prove that the midpoints of the segments $M_1M_2$ are on the same a straight line that does not depend on the position of points $T_1$, $T_2$.

A point $E$ lies on the extension of the side $AD$ of the rectangle $ABCD$ over $D$. The ray $EC$ meets the circumcircle $\omega$ of $ABE$ at the point $F\ne E$. The rays $DC$ and $AF$ meet at $P$. $H$ is the foot of the perpendicular drawn from $C$ to the line $\ell$ going through $E$ and parallel to $AF$. Prove that the line $PH$ is tangent to $\omega$. (A. Kuznetsov)

Given an isosceles triangle $ABC$ with a right angle at $C,$ construct the center $M$ and radius $r$ of a circle cutting on segments $AB, BC, CA$ the segments $DE, FG,$ and $HK,$ respectively, such that $\angle DME + \angle FMG + \angle HMK = 180^\circ$ and $DE : FG : HK = AB : BC : CA.$