Found problems: 85335
Let $n$ be a natural number. We define $f(n)$ as the number of ways to express $n$ as a sum of powers of $2$, where the order of the terms is taken into account. For example, $f(4) = 6$, because $4$ can be written as:
\begin{align*}
4;\\
2 + 2;\\
2 + 1 + 1;\\
1 + 2 + 1;\\
1 + 1 + 2;\\
1 + 1 + 1 + 1.
\end{align*}
Find the smallest $n$ greater than $2019$ for which $f(n)$ is odd.
Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality
$a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.
It is known that a $n$-vertex contains within itself a polyhedron $M$ with a center of symmetry at some point $Q$ and is itself contained in a polyhedron homothetic to $M$ with a homothety center at a point $Q$ and coefficient $k$. Find the smallest value of $k$ if
a) $n = 4$,
b) $n = 5$.
Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.
Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.
[i]Evan O'Dorney.[/i]
How many ordered pairs $\left(a,b\right)$ of positive integers are there such that \[\gcd\left(a,b\right)^3=\mathrm{lcm}\left(a,b\right)^2=4^6\] is true?
[i]2019 CCA Math Bonanza Individual Round #4[/i]
For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.
Let $k$ be the circumference of the triangle $ABC.$ The point $D$ is an arbitrary point on the segment $AB.$ Let $I$ and $J$ be the centers of the circles which are tangent to the side $AB,$ the segment $CD$ and the circle $k.$ We know that the points $A, B, I$ and $J$ are concyclic. The excircle of the triangle $ABC$ is tangent to the side $AB$ in the point $M.$ Prove that $M \equiv D.$
a) We are playing the following game on this table:
In each move we select a row or a column of the table, reduce two neighboring numbers in that row or column by $1$ and increase the third one by $1$. After some of these moves can we get to a table with all the same entries?
b) This time we have the choice to arrange the integers from $1$ to $9$ in the$ 3 \times3$ table. Still using the same moves now our aim is to create a table with all the same entries, maximising the value of the entries. What is the highest possible number we can achieve?
Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that
\[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\]
holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.
Kayla rolls four fair $6$-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$?
$\textbf{(A)}\frac{2}{3}~\textbf{(B)}\frac{19}{27}~\textbf{(C)}\frac{59}{81}~\textbf{(D)}\frac{61}{81}~\textbf{(E)}\frac{7}{9}$
There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends’ homes.
The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is
$ \textbf{(A)}\ 112 \qquad
\textbf{(B)}\ 100 \qquad
\textbf{(C)}\ 92 \qquad
\textbf{(D)}\ 88 \qquad
\textbf{(E)}\ 80$
Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$ where $a\in Z$, in which one of the numbers is equal to the sum of all the others.
(Bogdan Rublev)
Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.
Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that
\[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]
$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?
Let $ABCD$ a cyclic convex quadrilateral. There is a line $l$ parallel to $DC$ containing $A$. Let $P$ a point on $l$ closer to $A$ than to $B$. Let $B'$ the reflection of $B$ over the midpoint of $AD$. Prove that $\angle B'AP = \angle BAC$
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.
Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly.
a) Prove that $ AA_0,BB_0,CC_0$ are concurrent.
b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2\equal{}b^2\plus{}c^2$.
Let $ABC$ be a triangle in the plane with $AB = 13$, $BC = 14$, $AC = 15$. Let $M_n$ denote the smallest possible value of $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim_{n \to \infty} M_n$.
If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals:
$\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$
In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that
\[ \frac {E}{E_1} \geq 2.
\]
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.