Found problems: 85335
Areas of four surfaces of a tetrahedron are $S_1,S_2,S_3,S_4$. And the largest one of them is $S$. $\lambda=\frac{S_1+S_2+S_3+S_4}{S}$, then $\lambda$ always satisfies
$\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$
Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ($a_0\ne0$) such that $|f(x)|\le1$ for every $x\in[-1,1]$. Prove that
(a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which
(b) $|a_0|\le4$.
[i]V. Petkov[/i]
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
Given are $m\ge 3$ points in the plane. Prove that you can always choose three of these points $A,B,C$ such that
$$\angle ABC \le \frac{180^o}{m}.$$
[b]p1.[/b] $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins?
[b]p2.[/b] Compute ${150 \choose 20 }$ (mod $221$).
[b]p3.[/b] On the right triangle $ABC$, with right angle at$ B$, the altitude $BD$ is drawn. $E$ is drawn on $BC$ such that AE bisects angle $BAC$ and F is drawn on $AC$ such that $BF$ bisects angle $CBD$. Let the intersection of $AE$ and $BF$ be $G$. Given that $AB = 15$,$ BC = 20$, $AC = 25$, find $\frac{BG}{GF}$ .
[b]p4.[/b] What is the largest integer $n$ so that $\frac{n^2-2012}{n+7}$ is also an integer?
[b]p5.[/b] What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$?
[b]p6.[/b] Inside a LilacBall, you can find one of $7$ different notes, each equally likely. Delcatty must collect all $7$ notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so?
PS. You had better use hide for answers.
We have $4$ sticks. It is known, that for every $3$ sticks we can build a triangle with the same area. Is it true, that sticks have the same length?
Let be a sequence of functions $ \left( f_n \right)_{n\ge 2}:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ defined, for each $ n\ge 2, $ as
$$ f_n(x)=2nx^{2+n} -2(n+2)x^{1+n} +(2+n)x +1. $$
[b]a)[/b] Prove that $ f_n $ has an unique local maxima $ x_n, $ for any $ n\ge 2. $
[b]b)[/b] Show that $ 1=\lim_{n\to\infty } x_n. $
[i]Cătălin Zîrnă[/i]
We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?
Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by [i]Morteza Saghafian[/i]
A wall contains three switches $A,B,C$, each of which powers a light when flipped on. Every $20$ seconds, switch $A$ is turned on and then immediately turned off again. The same occurs for switch $B$ every $21$ seconds and switch $C$ every $22$ seconds. At time $t = 0$, all three switches are simultaneously on. Let $t = T > 0$ be the earliest time that all three switches are once again simultaneously on. Compute the number of times $t > 0$ before $T$ when at least two switches are simultaneously on.
Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?
[color=darkred]Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly decreasing function with $g(\mathbb{R})=(-\infty,0)$ . Prove that there are no continuous functions $f:\mathbb{R}\to\mathbb{R}$ with the property that there exists a natural number $k\ge 2$ so that : $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g$ . [/color]
Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$.
For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$
33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least $m$ stickers both don't have, is incomplete.
Determine if there exists a convex polyhedron such that
(1) it has 12 edges, 6 faces and 8 vertices;
(2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are pointing at each other?
[i]Proposed by David Altizio[/i]
Let \(ABC\) be an acute scalene triangle with orthocenter \(H\). Line \(BH\) intersects \(\overline{AC}\) at \(E\) and line \(CH\) intersects \(\overline{AB}\) at \(F\). Let \(X\) be the foot of the perpendicular from \(H\) to the line through \(A\) parallel to \(\overline{EF}\). Point \(B_1\) lies on line \(XF\) such that \(\overline{BB_1}\) is parallel to \(\overline{AC}\), and point \(C_1\) lies on line \(XE\) such that \(\overline{CC_1}\) is parallel to \(\overline{AB}\). Prove that points \(B\), \(C\), \(B_1\), \(C_1\) are concyclic.
[i]Proposed by Luke Robitaille[/i]
Prove that if \[ \sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)\] holds for a sequence $ \{a_n \}$ of nonnegative real numbers with some positive integer $ N$, then $ \alpha_{i+p} \geq p \alpha_i$ for $ i,p=1,2,...,$ where \[ \alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .\]
[i]L. Leindler[/i]
In rectangle $ABCD, AB= 5$ and $BC = 3$. Points $F$ and $G$ are on line segment $CD$ so that $DF = 1$ and $GC = 2$. Lines $AF$ and $BG$ intersect at $E$. What is the area of $\vartriangle AEB$?
$ABCDE$ is a cyclic convex pentagon, and $AC = BD = CE$. $AC$ and $BD$ intersect at $X$, and $BD$ and $CE$ intersect at $Y$ . If $AX = 6$, $XY = 4$, and $Y E = 7$, then the area of pentagon $ABCDE$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $ b$, $c$ are integers, $c$ is positive, $b$ is square-free, and gcd$(a, c) = 1$. Find $100a + 10b + c$.
Inside the acute-angled triangle $ABC$ we take $P$ and $Q$ two isogonal conjugate points. The perpendicular lines on the interior angle-bisector of $\angle BAC$ passing through $P$ and $Q$ intersect the segments $AC$ and $AB$ at the points $B_p\in AC$, $B_q\in AC$, $C_p\in AB$ and $C_q\in AB$, respectively. Let $W$ be the midpoint of the arc $BAC$ of the circle $(ABC)$. The line $WP$ intersects the circle $(ABC)$ again at $P_1$ and the line $WQ$ intersects the circle $(ABC)$ again at $Q_1$. Prove that the points $P_1$, $Q_1$, $B_p$, $B_q$, $C_p$ and $C_q$ lie on a circle.
[i]Proposed by P. Bibikov[/i]
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that
(a) $2t$ is [i]happy[/i],
(b) $3t$ is not [i]happy[/i]
Let $C$ be a right angle in triangle $ABC$. On legs $AC$ and$BC$ the squares $ACKL, BCMN$ are constructed outside of triangle. If $CE$ is an altitude of the triangle prove that $LEM$ is a right angle.