Found problems: 85335
Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.
Let $f(n)$ be the product of all factors of $n$. Find all natural numbers $n$ such that $f(n)$ is not a perfect power of $n$.
In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for
$a)$ $k=5;$
$b)$ $k=8.$
On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.
Let $p$ be a prime number and $a$ and $n$ positive nonzero integers. Prove that
if $2^p + 3^p = a^n$ then $n=1$
In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.
$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called [i]chordal[/i] if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be
$H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$
where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.
Given that $x > 1$, compute $x$ such that
$$\log_{16}(x) + \log_x(2)$$
is minimal.
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane.
Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$
[i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]
The expression $ 21x^2 \plus{} ax \plus{} 21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be one if $ a$ is:
$ \textbf{(A)}\ \text{any odd number} \qquad\textbf{(B)}\ \text{some odd number} \qquad\textbf{(C)}\ \text{any even number}$
$ \textbf{(D)}\ \text{some even number} \qquad\textbf{(E)}\ \text{zero}$
In $\triangle ABC$, angle bisector ıf $\widehat{CAB}$ meets $BC$ at $L$, angle bisector of $\widehat{ABC}$ meets $AC$ at $N$. Lines $AL$ and $BN$ meet at $O$. If $|NL| = \sqrt 3$, what is$|ON| + |OL|$?
$
\textbf{a)}\ 3\sqrt 3
\qquad\textbf{b)}\ 2\sqrt 3
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ 5
$
Let $(a_n)_{n\geq 1}$ be a positive real sequence given by $a_n=\sum \limits_{k=1}^n \frac{1}{k}$. Compute $$\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor$$where we denote by $\lfloor x\rfloor$ the integer part of $x$.
Among the spatial points $ A,B,C,D, $ at most two of are aparted at a distance greater than $ 1. $ Find the the maximum value of the expression:
$$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$
Bertalan thought about a $4$-digit positive number. Then he draw a simple graph on $4$ vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.
A cube consists of $4^3$ unit cubes each containing an integer. At each move, you choose a unit cube and increase by $1$ all the integers in the neighbouring cubes having a face in common with the chosen cube. Is it possible to reach a position where all the $4^3$ integers are divisible by $3,$ no matter what the starting position is?
Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.
For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_\text{four}) = 10 = 12_\text{eight}$, and $g(2020) = \text{the digit sum of } 12_\text{eight} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9.$ Find the remainder when $N$ is divided by $1000.$
Pick a $3$-digit number $abc$, which contains no $0$'s. The probability that this is a winning number is $\frac1a\cdot\frac1b\cdot\frac1c$. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways:
[list]
[*] For the lowest digit $n$ in the number, he rolls an $n$-sided die for each time that the digit appears, and gives the number $0$ probability of winning if an $n$ is rolled.
[*] For the largest digit $m$ in the number, he rolls an $m$-sided die once and scales the probability of winning by that die roll.
[/list]
If you choose optimally, what is the probability that your number is a winning number?
Find the minimum of $f(x,y,z)=x^3+12\frac{yz}{x}+16(\frac{1}{yz})^{\frac{3}{2}}$ where $x,y$, and $z$ are all positive.
Note: The problem as given in the tiebreaker did not specify that each of $x,y$, and $z$ had to be positive. Without this constraint, the answer is $-\infty$, as $x^3$ can be an arbitrarily large negative value and dominate the expression.
Let $f$ be a function from the set $Q$ of the rational numbers onto itself such that $f(x+y)=f(x)+f(y)+2547$ for all rational numbers $x,y$.
Moreover $f(2004) = 2547$.
Determine $f(2547).$
Pierre.
Let $a$ be any positive integer. Prove that there exists a unique pair of positive integers $x$ and $y$ such that
$$x +\frac12 (x + y - 1)(x + y- 2) = a.$$
$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $)
[i]Proposed by Mohsen Jamali[/i]
Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.