Found problems: 85335
Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$?
$ \textbf{(A)}\ -226 \qquad
\textbf{(B)}\ -144 \qquad
\textbf{(C)}\ -20 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 144$
There are $2015$ coins on a table. For $i = 1, 2, \dots , 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions:
(i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$;
(ii) $f(f(f(0)))=0$.
Prove that $f(0)=0$.
Let $ABC$ be a triangle and let $C$ be a circle that intersects the sides $BC, CA$ and $AB$ in the points $A_1, A_2, B_1, B_2$ and $C_1, C_2$ respectively. Prove that if $AA_1, BB_1$ and $CC_1$ are concurrent lines then $AA_2, BB_2$ and $CC_2$ are also concurrent lines.
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
A triangle $ABC$ has sides $BC = a, CA = b, AB = c$ with $a < b < c$ and area $S$. Determine the largest number $u$ and the least number $v$ such that, for every point $P$ inside $\triangle ABC$, the inequality $u \le PD + PE + PF \le v$ holds, where $D,E, F$ are the intersection points of $AP,BP,CP$ with the opposite sides.
A uniform ladder of mass $m$ and length $\mathcal{L}$ is resting on a wall. A man of mass $m$ climbs up the ladder and is in perfect equilibrium with the ladder when he is $\frac{2}{3}\mathcal{L}$ the way up the ladder. The ladder makes an angle of $ \theta = 30^\circ $ with the horizontal floor. If the coefficient of static friction between the ladder and the wall is the same as that between the ladder and the floor, which is $\mu$, what is $\mu$, expressed to the nearest thousandth?
[i](Proposed by Ahaan Rungta)[/i]
Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$
If $a>1,b>1$ are the legths of the catheti of an right triangle and $c$ the length of its hypotenuse, prove that
$a+b\leq c\sqrt 2$
Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.
Let $u\in C^2(\overline D)$, $u=0$ on $\partial D$ where $D$ is the open unit ball in $\mathbb R^3$. Prove that the following inequality holds for all $\varepsilon>0$:
$$\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.$$(We recall that $\nabla u$ and $\Delta u$ are the gradient and Laplacian, respectively.)
A circle centered at $O$ and inscribed in triangle $ABC$ meets sides $AC$;$AB$;$BC$ at $K$;$M$;$N$, respectively. The median $BB_1$ of the triangle meets $MN$ at $D$. Show that $O$;$D$;$K$ are collinear.
[i]M. Sonkin[/i]
Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
Dora has $8$ rods with lengths $1, 2, 3, 4, 5, 6, 7$ and $8$ cm. Dora chooses $4$ of the rods and uses them to assemble a trapezoid (the $4$ chosen rods must be the $4$ sides). How many different trapezoids can she obtain in this way?
Two trapezoids are considered different if they are not congruent.
Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.
Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define
$f(P)=PA*BC+PD*CA+PC*AB$.
(1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum.
(2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.
At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does the coo-coo bird coo-coo from $12:42$ PM on Monday until $3:42$ AM on Wednesday?
On the table there are $50$ stacks of coins that have $1,2,3,…,50$ coins respectively. Ana and Beto play the following game in turns:
First, Ana chooses one of the $50$ piles on the table, and Beto decides if that pile is for Ana or for him.
Then, Beto chooses one of the $49$ remaining piles on the table, and Ana decides if that pile is for her or for Beto.
They continue playing alternately in this way until one of the players has $25$ batteries.
When that happens, the other player takes all the remaining stacks on the table and whoever has the most coins wins.
Determine which of the two players has a winning strategy.
For a rectangle $R$ with integral side lengths, denote by $D(a, b)$ the number of ways of covering $R$ by congruent rectangles with integral side lengths formed by a family of cuts parallel to one side of $R$. Determine the perimeter $P$ of the rectangle $R$ for which $\frac{D(a,b)}{a+b}$ is maximal.
Given a circle with diameter $AB$. Points $C$ and $D$ are selected on the circumference of the circle such that the chord $CD$ intersects $AB$ inside the circle, at point $P$. The ratio of the arc length $\overarc {AC}$ to the arc length $\overarc {BD}$ is $4 : 1$ , while the ratio of the arc length $\overarc{AD}$ to the arc length $\overarc {BC}$ is $3 : 2$ . Find $\angle{APC}$ , in degrees.
Every day, Heesu talks to Sally with some probability $p$. One day, after not talking to Sally the previous day, Heesu resolves to ask Sally out on a date. From now on, each day, if Heesu has talked to Sally each of the past four days, then Heesu will ask Sally out on a date. Heesu’s friend remarked that at this rate, it would take Heesu an expected $2800$ days to finally ask Sally out. Suppose $p=\tfrac{m}{n}$, where $\gcd(m, n) = 1$ and $m, n > 0$. What is $m + n$?
Let $\{x_n\}$ be a real sequence defined by:
\[x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n\]
For all $n=1,2,3...$ and a is a real number.
Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.
Let $A > 1$ and $B > 1$ be real numbers and (xn) be a sequence of numbers in the interval $[1,AB]$. Prove that there exists a sequence $(y_n)$ of numbers in the interval $[1,A]$ such that
$$\frac{x_m}{x_n}\le B\frac{y_m}{y_n} \,\,\, for \,\,\, all \,\,\, m,n = 1,2,...$$
Let $a, b, c, d, e$ be real numbers satisfying the following conditions.
\[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.