Found problems: 85335
There are $2n$ distinct points on a circle. The numbers $1$ through $2n$ are randomly assigned to this one points distributed. Each point is connected to exactly one other point, so that no of the resulting connecting routes intersect. If a segment connects the numbers $a$ and $b$, so we assign the value $ |a - b|$ to the segment . Show that we can choose the routes such that the sum of these values results $n^2$.
Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.
The Fibonacci sequence is given by $a_1 = 1, a_2 = 2$ and $a_{n+1} = a_n +a_{n-1}$ for $n > 1$. Express $a_{2n}$ in terms of only $a_{n-1},a_n,a_{n+1}$.
Let $n$ be a positive integer. The following $35$ multiplication are performed:
$$1 \cdot n, 2 \cdot n, \dots, 35 \cdot n.$$
Show that in at least one of these results the digit $7$ appears at least once.
Dunno wrote several different natural numbers on the board and divided (in his head) the sum of these numbers by their product. After this, Dunno erased the smallest number and divided (again in his mind) the amount of the remaining numbers by their product. The second result was $3$ times greater than the first. What number did Dunno erase?
Lason Jiu gives a problem to Sick Nong and Ayush Agrawal. Sick takes 6 minutes to solve the problem, while Ayush takes 9 minutes. Sick has a 1/3 chance of solving correctly and Ayush has a 2/3 chance of solving correctly. If they solved it incorrectly, they resume solving with the same time and accuracy. Lason gives a rubber chicken to the first person who solves it correctly. If Sick and Ayush solve the question at the same time, Lason checks Sick's work first. The probability that Ayush wins the rubber chicken can be expressed as $\frac{p}{q}$. Find $p+q$.
[i]2022 CCA Math Bonanza Individual Round #8[/i]
$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If
$d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $
find all possible values of $d$.
The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $.
Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that
\[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\]
for every positive integer $k$.
[i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]
Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$
Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]
$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.
Let $ ABC$ be a triangle with $ AB \equal{} AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K \equal{} 45^\circ$ . Find all possible values of $ \angle C AB$ .
[i]Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea [/i]
Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$.
[i]Ankan Bhattacharya and Evan Chen[/i]
How many positive integers are there such that $\left \lfloor \frac m{11} \right \rfloor = \left \lfloor \frac m{10} \right \rfloor$? ($\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$.)
$
\textbf{(A)}\ 44
\qquad\textbf{(B)}\ 48
\qquad\textbf{(C)}\ 52
\qquad\textbf{(D)}\ 54
\qquad\textbf{(E)}\ 56
$
Let $k_1$ and $k_2$ be two circles that are externally tangent at point $C$. We have a point $A$ on $k_1$ and a point $B$ on $k_2$ such that $C$ is an interior point of segment $AB$. Let $k_3$ be a circle that passes through points $A$ and $B$ and intersects circles $k_1$ and $k_2$ another time at points $M$ and $N$ respectively. Let $k_4$ be the circumscribed circle of triangle $CMN$. Prove that the centres of circles $k_1, k_2, k_3$ and $k_4$ all lie on the same circle.
The image that maps $x$ to $1 - x$ is called [i]complement[/i], the image that maps $x$ to $\frac{1}{x}$ is called [i]invert[/i]. Two numbers $x$ and $y$ are called related if they can be transferred into each other by means of [i]complementation [/i]and/or [i]inversion[/i]. A [i]family [/i] is a collection of numbers where every two elements are related. Determine the maximum size $n$ of such a family.
Show that the number line can be divided into $n$ parts, such that each of those $n$ parts contains exactly one number from each $n$-number family.
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$, $BD$ of $ABCD$ meet at $P$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ($O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$.
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.
Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold:
$(i)$ $f(f(a)+b) \mid b^a-1$;
$(ii)$ $f(f(a))\geq f(a)-1$.
Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.
Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
[i]Proposed by Hojoo Lee, Korea[/i]
Source: 2018 Canadian Open Math Challenge Part A Problem 3
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Points $(0,0)$ and $(3\sqrt7,7\sqrt3)$ are the endpoints of a diameter of circle $\Gamma.$ Determine the other $x$ intercept of $\Gamma.$
Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$
[b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$