Found problems: 85335
For each $t \geq 0$ let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which
$$f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)).$$
Define necessary and sufficient conditions on $ t$ for $S_t $ to be closed under multiplication.
A list of 9 real numbers consists of $1$, $2.2 $, $3.2 $, $5.2 $, $6.2 $, $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) infinitely many}\qquad
$
Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$.
Compute the last two digits of $$9^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}$$
Let $f(x) = x^3 + 3x - 1$ have roots $ a, b, c$. Given that $\frac{1}{a^3 + b^3}+\frac{1}{b^3 + c^3}+\frac{1}{c^3 + a^3}$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $gcd(m, n) = 1$, find $100m + n$.
In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?
Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$
A school security guard works from Monday to Saturday from $7:30 am$ to $12:00 pm$ ($7:30$ to $12:00$). He also works the night shift, from Monday to Friday from $6pm$to $10pm$ ($18:00$ to $22:00$) . He receives $75MT$ per hour, up to $40$ hours of work per week. For the remaining hours of weekly work, he receives $95MT$ per hour. So, considering that a month has four weeks, what will be this security guard's monthly salary?
Alice and Bob play the following game: starting with the number $2$ written on a blackboard, each player in turn changes the current number $n$ to a number $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{22...2}_{2020}$. Assuming perfect play, who will win the game.
Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that for all $x, y \in \mathbb R$,
$$f(f(x) + y)^2 = (x-y)(f(x) - f(y)) + 4f(x) f(y).$$
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that
\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
$n$ players are playing a game. Each player has $n$ tokens. Every turn, two players with at least one token are randomly selected. The player with less tokens gives one token to the player with more tokens. If both players have the same number of tokens, a coin flip decides which player receives a token and which player gives a token. The game ends when one player has all the tokens. If $n = 2019$, suppose the maximum number of turns the game could take to end can be written as $\frac{1}{d} (a \cdot 2019^3 + b \cdot 2019^2 + c \cdot 2019)$ for integers $a, b, c, d$. Find $\frac{abc}{d}$
.
Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively.
a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$.
b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?
Determine the positive integers $n \ge 3$ such that, for every integer $m \ge 0$, there exist integers $a_1, a_2,..., a_n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_1a_2 + a_2a_3 + ...+a_{n-1}a_n + a_na_1 = -m$
Alexandru Mihalcu
An $8 \times 8$ chessboard consists of $64$ square units. In some of the unit squares of the board, diagonals are drawn so that any two diagonals have no common points. What is the maximum number of diagonals that can be drawn?
Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).
Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.
Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$, any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$.
In quadrilateral $ABCD$, $AC = 5$, $CD = 7$, and $AD = 3$. The angle bisector of $\angle CAD$ intersects $CD$ at $E$. If $\angle CBD = 60^o$ and $\angle AED = \angle BEC$, compute the value of $AE + BE$.
Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?
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$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) }8$
Let $m$ be a given positive integer. Define $a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.$ Prove that there are infinite many integers and infinite many non-integers in the sequence $\{a_k\}$.
Find the number of integers $x$ such that the following three conditions all hold:
$\bullet$ $x$ is a multiple of $5$
$\bullet$ $121 < x < 1331$
$\bullet$ When $x$ is written as an integer in base $11$ with no leading $0$s (i.e. no $0$s at the very left), its rightmost digit is strictly greater than its leftmost digit.
You plan to organize your birthday party, which will be attended either by exactly $m$ persons or by exactly $n$ persons (you are not sure at the moment). You have a big birthday cake and you want to divide it into several parts (not necessarily equal), so that you are able to distribute the whole cake among the people attending the party with everybody getting cake of equal mass (however, one may get one big slice, while others several small slices - the sizes of slices may differ). What is the minimal number of parts you need to divide the cake, so that it is possible, regardless of the number of guests.
Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.