Found problems: 85335
2011 F = Ma, 24
A turntable is supported on a Teflon ring of inner radius $R$ and outer radius $R+\sigma$ ($\sigma<<R$), as shown in the diagram. To rotate the turntable at a constant rate, power must be supplied to overcome friction. The manufacturer of the turntable wishes to reduce the power required without changing the rotation rate, the weight of the turntable, or the coefficient of friction of the Teflon surface. Engineers propose two solutions: increasing the width of the bearing (increasing $\sigma$), or increasing the radius (increasing $R$). What are the effects of these proposed changes?
[asy]
size(200);
draw(circle((0,0),5.5),linewidth(2));
draw(circle((0,0),7),linewidth(2));
path arrow1 = (0,0)--5*dir(50);
draw(arrow1,EndArrow);
label("R",arrow1,NW);
draw((3,0)--(5.5,0),EndArrow);
path arrow2 = ((10,0)--(7,0));
draw(arrow2,EndArrow);
label("$\delta$",arrow2,N);
[/asy]
(A) Increasing $\sigma$ has no significant effect on the required power; increasing $R$ increases the required power.
(B) Increasing $\sigma$ has no significant effect on the required power; increasing $R$ decreases the required power.
(C) Increasing $\sigma$ increases the required power; increasing $R$ has no significant effect on the required power.
(D) Increasing $\sigma$ decreases the required power; increasing $R$ has no significant effect on the required power.
(E) Neither change has a significant effect on the required power.
2004 Korea - Final Round, 1
On a circle there are $n$ points such that every point has a distinct number. Determine the number of ways of choosing $k$ points such that for any point there are at least 3 points between this point and the nearest point. (clockwise) ($n,k\geq 2$)
2013 ELMO Shortlist, 7
A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$).
What is the maximum possible number of filled black squares?
[i]Proposed by David Yang[/i]
2017 Pan African, Problem 5
The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game :
each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?
2019 Saudi Arabia Pre-TST + Training Tests, 2.3
Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.
2014 CentroAmerican, 1
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
2023 Romania National Olympiad, 3
We consider triangle $ABC$ with $\angle BAC = 90^{\circ}$ and $\angle ABC = 60^{\circ}.$ Let $ D \in (AC) , E \in (AB),$ such that $CD = 2 \cdot DA$ and $DE $ is bisector of $\angle ADB.$ Denote by $M$ the intersection of $CE$ and $BD$, and by $P$ the intersection of $DE$ and $AM$.
a) Show that $AM \perp BD$.
b) Show that $3 \cdot PB = 2 \cdot CM$.
2012 Mexico National Olympiad, 4
The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the [i]house[/i] of $n$.
How many integers less than $26000$ share the same [i]house[/i] as $2012$?
1975 AMC 12/AHSME, 8
If the statement "All shirts in this store are on sale." is false, then which of the following statements must be true?
I. All shirts in this store are at non-sale prices.
II. There is some shirt in this store not on sale.
III. No shirt in this store is on sale.
IV. Not all shirts in this store are on sale.
$ \textbf{(A)}\ \text{II only} \qquad
\textbf{(B)}\ \text{IV only} \qquad
\textbf{(C)}\ \text{I and III only} \qquad$
$ \textbf{(D)}\ \text{II and IV only} \qquad
\textbf{(E)}\ \text{I, II and IV only}$
1995 Brazil National Olympiad, 1
$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$ coincide, then it must be a square.
1941 Moscow Mathematical Olympiad, 086
Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.
2005 Indonesia MO, 6
Find all triples $ (x,y,z)$ of integers which satisfy
$ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$
$ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$
$ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 5
Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers.
Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$
2014 Balkan MO Shortlist, G4
Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.
2008 Portugal MO, 6
Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.
1998 Iran MO (3rd Round), 3
Let $n(r)$ be the maximum possible number of points with integer coordinates on a circle with radius $r$ in Cartesian plane. Prove that $n(r) < 6\sqrt[3]{3 \pi r^2}.$
2017 CCA Math Bonanza, L2.2
Non-degenerate triangle $ABC$ has $AB=20$, $AC=17$, and $BC=n$, an integer. How many possible values of $n$ are there?
[i]2017 CCA Math Bonanza Lightning Round #2.2[/i]
Indonesia Regional MO OSP SMA - geometry, 2018.3
Let $ \Gamma_1$ and $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $P$. The line $\ell$ passing through $O_1$ is tangent to $\Gamma_2$ at point $A$. The line $\ell$ intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$.
1991 IMO Shortlist, 10
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
[b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
Putnam 1938, A2
A solid has a cylindrical middle with a conical cap at each end. The height of each cap equals the length of the middle. For a given surface area, what shape maximizes the volume?
2017 Peru Iberoamerican Team Selection Test, P5
Let $ABCD$ be a trapezoid of bases $AD$ and $BC$ , with $AD> BC$, whose diagonals are cut at point $E$. Let $P$ and $Q$ be the feet of the perpendicular drawn from $E$ on the sides $AD$ and $BC$, respectively, with $P$ and $Q$ in segments $AD$ and $BC,$ respectively. Let $I$ be the center of the triangle $AED$ and let $K$ be the point of intersection of the lines $AI$ and $CD$. If $AP + AE = BQ + BE$, show that $AI = IK$.
1969 AMC 12/AHSME, 26
[asy]
size(180);
defaultpen(linewidth(0.8));
real r=4/5;
draw((-1,0)..(-6/7,r/3)..(0,r)..(6/7,r/3)..(1,0),linetype("4 4"));
draw((-1,0)--(1,0)^^origin--(0,r));
label("$A$",(-1,0),W);
label("$B$",(1,0),E);
label("$M$",origin,S);
label("$C$",(0,r),N);
[/asy]
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is:
$\textbf{(A) }1\qquad
\textbf{(B) }15\qquad
\textbf{(C) }15\tfrac13\qquad
\textbf{(D) }15\tfrac12\qquad
\textbf{(E) }15\tfrac34$
2009 Germany Team Selection Test, 1
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2013 USAMTS Problems, 5
Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?
2023 Malaysian IMO Team Selection Test, 5
Let $ABCD$ be a cyclic quadrilateral, with circumcircle $\omega$ and circumcenter $O$. Let $AB$ intersect $CD$ at $E$, $AD$ intersect $BC$ at $F$, and $AC$ intersect $BD$ at $G$.
The points $A_1, B_1, C_1, D_1$ are chosen on rays $GA$, $GB$, $GC$, $GD$ such that:
$\bullet$ $\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}$
$\bullet$ The points $A_1, B_1, C_1, D_1, O$ lie on a circle.
Let $A_1B_1$ intersect $C_1D_1$ at $K$, and $A_1D_1$ intersect $B_1C_1$ at $L$. Prove that the image of the circle $(A_1B_1C_1D_1)$ under inversion about $\omega$ is a line passing through the midpoints of $KE$ and $LF$.
[i]Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin[/i]