Found problems: 85335
Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
[i]Ray Li[/i]
Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality
$$ f\left( x+x_n \right)\geqslant f(x) $$
is true.
[b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality
$$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$
is true.
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
Let $P$ be a point on circumcircle of triangle $ABC$ on arc $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$ intersect at point $E$, and lines $AC$ and $BP$ intersect at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that:
${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$
Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$.
If there is a real number $C$, such that
\[ (i-j)m_k+(j-k)m_i+(k-i)m_j=C\]
for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.
Two circles $C_{1}$ and $C_{2}$ intersect at points $P$ and $Q$. Their common tangent, closer to $P$ than to $Q$, touches $C_{1}$ at $A$ and $C_{2}$ at $B$. The tangents to $C_{1}$ and $C_{2}$ at $P$ meet the other circle at points $E \not = P$ and $F \not = P$ , respectively. Let $H$ and $K$ be the points on the rays $AF$ and $BE$ respectively such that $AH = AP$ and $BK = BP$ . Prove that $A,H,Q,K,B$ lie on a circle.
Find the smallest real number $p(\leq 1)$ that satisfies the following condition.
(Condition) For real numbers $x_1, x_2, \dots, x_{2024}, y_1, y_2, \dots, y_{2024}$, if
[list]
[*] $0 \leq x_1 \leq x_2 \leq \dots \leq x_{2024} \leq 1$,
[*] $0 \leq y_1 \leq y_2 \leq \dots \leq y_{2024} \leq 1$,
[*] $\displaystyle \sum_{i=1}^{2024}x_i = \displaystyle \sum_{i=1}^{2024}y_i = 2024p$,
[/list]
then the inequality $\displaystyle \sum_{i=1}^{2024}x_i(y_{2025-i}-y_{2024-i}) \geq 1 - p$ holds.
Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them.
[i]Proposed by [b]vvluo[/b] & [b]richy[/b][/i]
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.
Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly$ 20$ di erent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$. Find all possible values of $N$.(Both $1$ and $m$ are counted as divisors of the number $m$.)
Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
Alice writes differents real numbers in the board, if $a,b,c$ are three numbers in this board, least one of this numbers $a + b, b + c, a + c$ also is a number in the board. What's the largest quantity of numbers written in the board???
Find all pairs $(p,q)$ of prime numbers such that
$$p^q-4~|~q^p-1.$$
[i](Vlad Robu)[/i]
What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?
In isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\vartriangle ACM$ with the circle with center $B$ passing through $M$, such that $P \ne M$. If $\angle BPC = 135^o$, then $\frac{CP}{AP}$ can be written as $a +\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime. Find $a + b$.
Petya digs the garden bed alone for $a$ minutes longer than he does with Vasya. Vasya digs up the same bed for $b$ minutes longer than he would have done with Petya. How many minutes does it take Vasya and Petya to dig up the same bed together? orthogonal).
Find the largest interval $ M \subseteq \mathbb{R^ \plus{} }$, such that for all $ a$, $ b$, $ c$, $ d \in M$ the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}\]
holds. Does the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}\]
hold too for all $ a$, $ b$, $ c$, $ d \in M$?
($ \mathbb{R^ \plus{} }$ denotes the set of positive reals.)
We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.
Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define $f_j(r)$ and $g_j(r)$ by
\[f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\rceil, n) + \min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),\]
where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that
\[\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)\]
for all positive real numbers $r$.
On Day $1$, Alice starts with the number $a_1=5$. For all positive integers $n>1$, on Day $n$, Alice randomly selects a positive integer $a_n$ between $a_{n-1}$ and $2a_{n-1}$, inclusive. Given that the probability that all of $a_2,a_3,\ldots,a_7$ are odd can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.
[i]2021 CCA Math Bonanza Lightning Round #1.4[/i]
There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.
[i]Proposed by Peter Novotný, Slovakia[/i]
Find all $3$-digit positive integers that are $34$ times the sum of their digits.
An $\textrm{alien}$ script has $n$ letters $b_1,b_2,\dots,b_n$. For some $k<n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered $\textbf{sacred}$ if:
i. no letter appears twice and,
ii. if a letter $b_i$ appears in the word then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$).
For example, if $n = 7$ and $k = 3$ then $b_1b_3b_6, b_3b_1b_6, b_2b_4b_6$ are sacred $3$-words. On the other hand $b_1b_7b_4, b_2b_2b_6$ are not sacred.
What is the total number of sacred $k$-words?
Use your formula to find the answer for $n = 10$ and $k = 4$.
Find all solutions of the equation
$$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$
A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?