Found problems: 85335
Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties:
$\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.
$\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.
A sequence $(a_k)_{k=1}^{\infty}$ has the property that there is a natural number $n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_{n+k} = a_k$ for all $k$. Prove that there exists a natural number $N$ such that
$$\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...$$
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property:
\[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\]
[i]Proposed by Nikolay Nikolov[/i]
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$
Richard and Kaarel are taking turns to choose numbers from the set $\{1,\dots,p-1\}$ where $p > 3$ is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel.
Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by $p$. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by $p$. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?
What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
[list]
[*] $D(1) = 0$;
[*] $D(p)=1$ for all primes $p$;
[*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$.
[/list]
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.
Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$.
For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
Find the sum of the two smallest odd primes $p$ such that for some integers $a$ and $b$, $p$ does not divide $b$, $b$ is even, and $p^2=a^3+b^2$.
[i]2021 CCA Math Bonanza Individual Round #13[/i]
Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.
If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor +\dots \lfloor \frac{2^{2009}}{5} \rfloor$, what is the remainder when $2^{2010}$ is divided by $N$?
$ \textbf{(A)}\ 5034
\qquad\textbf{(B)}\ 5032
\qquad\textbf{(C)}\ 5031
\qquad\textbf{(D)}\ 5028
\qquad\textbf{(E)}\ 5024
$
In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+y+2z+3w=n-1$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that
(i). $a+b+c+d=n$.
(ii). $a \ge b$, $c \ge d$, $a \ge d$.
(iii). $b < c$.
Prove that for all $n$, $p(n) = q(n)$.
Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$
We will call one of the cells of a rectangle 11 x 13 “[i]peculiar[/i]” , if after removing it the remaining figure can be cut into squares 2 x 2 and 3 x 3. How many of the 143 cells are “[i]peculiar[/i]”?
In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.
Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following:
[i](i) [/i] $|AB| \leq a$ ($a$ fixed);
[i](ii) [/i] $|EF| = l$ ($l$ fixed);
[i](iii)[/i] the sum of squares of the nonparallel sides of the trapezoid is constant.
[hide="Remark"]
[b]Remark.[/b] The constants are chosen so that such trapezoids exist.[/hide]
Find all ordered pairs of positive integers $(m,n)$ such that :
$125*2^n-3^m=271$
Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$. Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$.
Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things:
1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa)
2. Change the first bit after the first $1$.
Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.
Let $a_1,a_2,\ldots,a_k$ be real numbers and $a_1+a_2+\ldots+a_k=0$. Prove that
\[ \max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right). \]
Let \( p_1, p_2, \cdots, p_{2025} \) be real numbers. For \( 1 \leq i \leq 2025 \), let
\[\{a_n^{(i)}\}_{n \geq 0}\]
be an infinite real sequence satisfying
\[a_0^{(i)} = 0.\]
It is known that:
(1)
\[a_1^{(1)}, a_1^{(2)}, \cdots, a_1^{(2025)}\]
are not all zero.
(2) For any integer \( n \geq 0 \) and any \( 1 \leq i \leq 2025 \), the following holds:
\[p_i \cdot a_n^{(i+1)} = a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)},\]
where the sequence
\[\{a_n^{(2026)}\}\]
satisfies
\[a_n^{(2026)} = a_n^{(1)}, \, n = 0, 1, 2, \cdots.\]
Prove that there exists a positive real number \( r \) such that for infinitely many positive integers \( n \),
\[\max \left\{ |a_n^{(1)}|, |a_n^{(2)}|, \cdots, |a_n^{(2025)}|\right\} \geq r.\]
When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?
$\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$
Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point.
[b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective.
[b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$.
(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$.
(b) Show that $ a_{2008} \neq 0$