Found problems: 26
1990 Putnam, B6
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
2002 SNSB Admission, 3
Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.
1998 AMC 8, 17
Problems 15, 16, and 17 all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.
17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?
$ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $
2012 AIME Problems, 5
Let $B$ be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer 1 is obtained.
2007 F = Ma, 12
A $2$-kg rock is suspended by a massless string from one end of a uniform $1$-meter measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the $0.20$-meter mark?
[asy]
size(250);
draw((0,0)--(7.5,0)--(7.5,0.2)--(0,0.2)--cycle);
draw((1.5,0)--(1.5,0.2));
draw((3,0)--(3,0.2));
draw((4.5,0)--(4.5,0.2));
draw((6,0)--(6,0.2));
filldraw((1.5,0)--(1.2,-1.25)--(1.8,-1.25)--cycle, gray(.8));
draw((0,0)--(0,-0.4));
filldraw((0,-0.4)--(-0.05,-0.4)--(-0.1,-0.375)--(-0.2,-0.375)--(-0.3,-0.4)--(-0.3,-0.45)--(-0.4,-0.6)--(-0.35,-0.7)--(-0.15,-0.75)--(-0.1,-0.825)--(0.1,-0.84)--(0.15,-0.8)--(0.15,-0.75)--(0.25,-0.7)--(0.25,-0.55)--(0.2,-0.4)--(0.1,-0.35)--cycle, gray(.4));
[/asy]
$ \textbf {(A) } 0.20 \, \text{kg} \qquad \textbf {(B) } 1.00 \, \text{kg} \qquad \textbf {(C) } 1.33 \, \text{kg} \qquad \textbf {(D) } 2.00 \, \text{kg} \qquad \textbf {(E) } 3.00 \, \text{kg} $
2015 Romania Team Selection Tests, 2
Let $ABC$ be a triangle . Let $A'$ be the center of the circle through the midpoint of the side $BC$ and the orthogonal projections of $B$ and $C$ on the lines of support of the internal bisectrices of the angles $ACB$ and $ABC$ , respectively ; the points $B'$ and $C'$ are defined similarly . Prove that the nine-point circle of the triangle $ABC$ and the circumcircle of $A'B'C'$ are concentric.
2008 ITest, 88
A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?
1998 All-Russian Olympiad, 1
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
2013 AMC 12/AHSME, 21
Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $?
$ \textbf{(A)} \ (\log 2016, \log 2017) $
$ \textbf{(B)} \ (\log 2017, \log 2018) $
$ \textbf{(C)} \ (\log 2018, \log 2019) $
$ \textbf{(D)} \ (\log 2019, \log 2020) $
$ \textbf{(E)} \ (\log 2020, \log 2021) $
2013 Purple Comet Problems, 18
Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.
1999 AIME Problems, 8
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
1983 AIME Problems, 5
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?
1981 Miklós Schweitzer, 10
Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$.
[i]T. F. Mori, G. J. Szekely[/i]
2007 F = Ma, 11
A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest.
[asy]
size(225);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((0,0),1),gray(.7));
draw((0,-1)--(2,-1),EndArrow);
label("$\vec{F}$",(1, -1),S);
label("Disk",(-1,0),W);
filldraw(circle((5,0),1),gray(.7));
filldraw(circle((5,0),0.75),white);
draw((5,-1)--(7,-1),EndArrow);
label("$\vec{F}$",(6, -1),S);
label("Hoop",(6,0),E);
filldraw(circle((10,0),1),gray(.5));
draw((10,-1)--(12,-1),EndArrow);
label("$\vec{F}$",(11, -1),S);
label("Sphere",(11,0),E);
[/asy]
$ \textbf{(A)} \ \text{disk, hoop, sphere}$
$\textbf{(B)}\ \text{sphere, disk, hoop}$
$\textbf{(C)}\ \text{hoop, sphere, disk}$
$\textbf{(D)}\ \text{disk, sphere, hoop}$
$\textbf{(E)}\ \text{hoop, disk, sphere} $
2016 All-Russian Olympiad, 1
A carpet dealer,who has a lot of carpets in the market,is available to exchange a carpet of dimensions $a\cdot b$ either with a carpet with dimensions $\frac{1}{a}\cdot \frac{1}{b}$ or with two carpets with dimensions $c\cdot b$ and $\frac{a}{c}\cdot b$ (the customer can select the number $c$).The dealer supports that,at the beginning he had a carpet with dimensions greater than $1$ and,after some exchanges like the ones we described above,he ended up with a set of carpets,each one having one dimension greater than $1$ and one smaller than $1$.Is this possible?
[i]Note:The customer can demand from the dealer to consider a carpet of dimensions $a\cdot b$ as one with dimensions $b\cdot a$.[/i]
2002 AIME Problems, 11
Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8,$ and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p},$ where $m,$ $n,$ and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p.$
2013 Putnam, 6
Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$
\[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\
&w(a,b)&-2&-1&0&1&2\\ \hline
&-2&-1&-2&2&-2&-1\\
&-1&-2&4&-4&4&-2\\
a&0&2&-4&12&-4&2\\
&1&-2&4&-4&4&-2\\
&2&-1&-2&2&-2&-1\\ \hline\end{array}\]
For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)
2005 Gheorghe Vranceanu, 2
Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions:
[b]a)[/b] $ f''>0\implies f-\alpha >0 $
[b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $
2011 Putnam, B5
Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$
\[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\]
Prove there is a constant $B>0$ such that for all $n,$
\[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]
2011 USAMTS Problems, 4
Renata the robot packs boxes in a warehouse. Each box is a cube of side length $1$ foot. The warehouse floor is a square, $12$ feet on each side, and is divided into a $12$-by-$12$ grid of square tiles $1$ foot on a side. Each tile can either support one box or be empty. The warehouse has exactly one door, which opens onto one of the corner tiles.
Renata fits on a tile and can roll between tiles that share a side. To access a box, Renata must be able to roll along a path of empty tiles starting at the door and ending at a tile sharing a side with that box.
[list=a]
[*]Show how Renata can pack $91$ boxes into the warehouse and still be able to access any box.
[*]Show that Renata [b]cannot[/b] pack $95$ boxes into the warehouse and still be able to access any box.[/list]
2010 Laurențiu Panaitopol, Tulcea, 3
Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that:
$ \text{(i) supp} f''=f\left(\mathbb{R}\right) $
$ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $
Prove that:
[b]a)[/b] any such $ g $ is injective.
[b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $
$$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$
[i]Laurențiu Panaitopol[/i]
2008 Putnam, B2
Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$
2015 AMC 12/AHSME, 9
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color?
$\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$
2004 Iran MO (3rd Round), 27
$ \Delta_1,\ldots,\Delta_n$ are $ n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.
2012 Pre-Preparation Course Examination, 3
Consider the set
$\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$.
Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?