Found problems: 85335
Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis.
Find the ratio $\frac{V_2}{V_1}.$
Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n \minus{} 1}A_n$ $ (A_n \equal{} A_0)$ where the side $ A_{i \minus{} 1}A_i \equal{} i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i \plus{} 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum
$ \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i$
of these $ n$ angles.
In the figure below, $AB = 15$, $BD = 18$, $AF = 15$, $DF = 12$, $BE = 24$, and $CF = 17$. Find $BG : FG$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/dc171c52961442f9846d2fce858937ff9fb7e8.png[/img]
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
For each positive integer $n\geq 2$ we denote with $p(n)$ the largest prime number less than or equal to $n$, and with $q(n)$ the smallest prime number larger than $n$. Prove that \[ \sum^n_{k=2} \frac 1{p(k)q(k)} < \frac 12. \]
An ant starts at corner $A$ of a square room $ABCD$ with side length $2\sqrt{2}$. In the middle of the room, there is a circular pillar of radius $1$ centered at the center of $ABCD$. What is the minimum distance it has to travel to get to corner $C$?
Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$
[b][color=#f00]Same exact problem as 2010 Spring LMT Theme Round Problem 7?[/color][/b] :what?:
Let be given a tetrahedron $ ABCD$ such that triangle $ BCD$ equilateral and $ AB \equal{} AC \equal{} AD$. The height is $ h$ and the angle between two planes $ ABC$ and $ BCD$ is $ \alpha$. The point $ X$ is taken on $ AB$ such that the plane $ XCD$ is perpendicular to $ AB$. Find the volume of the tetrahedron $ XBCD$.
The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$.
What is the maximum possible cardinality of $M$?
[hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlist N2 (the one with Bertrand), Problem 5 is IMO Shortlist 2021 A1 and Problem 6 is USAMO 2002/1. Hence neither of these will be posted here. [/hide]
$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.
In quadrilateral $ABCD$, $m\angle B=m\angle C=120^\circ$, $AB=3$, $BC=4$, and $CD=5$. Find the area of $ABCD$.
$\textbf{(A) }15\qquad\textbf{(B) }9\sqrt3\qquad\textbf{(C) }\dfrac{45\sqrt3}4\qquad\textbf{(D) }\dfrac{47\sqrt3}4\qquad\textbf{(E) }15\sqrt3$
A trapezoid $ABCD$ is bicentral. The vertex $A$, the incenter $I$, the circumcircle $\omega$ and its center $O$ are given and the trapezoid is erased. Restore it using only a ruler.
Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?
Which of these numbers is less than its reciprocal?
$\textbf{(A)}\ -2\qquad
\textbf{(B)}\ -1\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 1\qquad
\textbf{(E)}\ 2$
A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.
Let a sequence $(x_n)$ be given and let $y_n = x_{n-1} +2 x_n $ for $n>1.$ Suppose that the sequence $(y_n)$ converges. Prove that the sequence $(x_n)$ converges, too.
Let $ABCD$ be a trapezoid of parallel bases $ BC$ and $AD$. If $\angle CAD = 2\angle CAB, BC = CD$ and $AC = AD$, determine all the possible values of the measure of the angle $\angle CAB$.
Ninety-one students in a school are distributed in three classes. Each student took part in a competition. It is known that among any six students of the same sex some two got the same number of points. Show that here are four students of the same sex who are in the same class and who got the same number of points.
Let circles $s_1$ and $s_2$ meet at points $A$ and $B$. Consider all lines passing through $A$ and meeting the circles for the second time at points $P_1$ and $P_2$ respectively. Construct by a compass and a ruler a line such that $AP_1.AP_2$ is maximal.
On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles .
[img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]
For positive integer $a,b,c,d$ there are $a+b+c+d$ points on plane which none of three are collinear. Prove there exist two lines $l_1, l_2 $ such that
(1) $l_1, l_2 $ are not parallel.
(2) $l_1, l_2 $ do not pass through any of $a+b+c+d$ points.
(3) There are $ a, b, c, d $ points on each region separated by two lines $l_1, l_2 $.
Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?
Let $M, N, P$ be the midpoints of the sides $AB, BC, CA$ of a triangle $ABC$. Let $X$ be a fixed point inside the triangle $MNP$. The lines $L_1, L_2, L_3$ that pass through point $X$ are such that $L_1$ intersects segment $AB$ at point $C_1$ and segment $AC$ at point $B_2$; $L_2$ intersects segment $BC$ at point $A_1$ and segment $BA$ at point $C_2$; $L_3$ intersects segment $CA$ at point $B_1$ and segment $CB$ at point $A_2$. Indicates how to construct the lines $L_1, L_2, L_3$ in such a way that the sum of the areas of the triangles $A_1A_2X, B_1B_2X$ and $C_1C_2X$ is a minimum.