Found problems: 85335
The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.
[i]UK - David Monk[/i]
Equilateral triangle $OAB$ of side length $1$ lies in the $xy$-plane ($O$ is the origin). Let $\ell, m$ be the vertical lines passing through $A,B$, respectively. Let $P,Q$ be on $\ell, m$ respectively such that the ratio $\overline{OP} : \overline{OQ} : \overline{PQ} = 3 : 3 : 5$. Let $Q = (x, y, z)$. If $z^2 = \frac{p}{q}$ . where $p, q$ are relatively prime positive integers, find $p + q$.
Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$.
Let $ABC$ be a triangle and let $M,N,P$ be points on the line $BC$ such that $AM,AN,AP$ are the altitude, the angle bisector and the median of the triangle, respectively. It is known that
$\frac{[AMP]}{[ABC]}=\frac{1}{4}$ and $\frac{[ANP]}{[ABC]}=1-\frac{\sqrt{3}}{2}$.
Find the angles of triangle $ABC$.
Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$, satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors.
Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.
The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.)
(AY Belov)
$ABCD$ - convex quadrilateral. $P$ is such point on $AC$ and inside $\triangle ABD$, that $$\angle ACD+\angle BDP = \angle ACB+ \angle DBP = 90-\angle BAD$$.
Prove that $\angle BAD+ \angle BCD =90$ or $\angle BDA + \angle CAB = 90$
Find all function $ f: \mathbb R\to \mathbb R$ satisfying the condition:
\[ f(f(x \minus{} y)) \equal{} f(x)\cdot f(y) \minus{} f(x) \plus{} f(y) \minus{} xy
\]
Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120.000$ Rs. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9.408$ Rs. How many centimeters long is the ruler used by the seller?
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
Eddy owns $5$ different cats, and has $9$ fish to distribute among the cats. Each cat gets at least $1$ fish and at most $3$ fish. If the fish are indistinguishable, how many ways can Eddy distribute the $9$ fish among the $5$ cats?
Determine all real $a$, such that the solutions to the system of equations
$\begin{cases}
\frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\
(2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a
\end{cases}$
form an interval with length $\frac{32}{225}$.
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 5
The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.
Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$
When does the equality hold?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Prove that the equation $ \frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c}\plus{}\frac{1}{abc} \equal{} \frac {12}{a \plus{} b \plus{} c}$ has infinitely many solutions $ (a,b,c)$ in natural numbers.
A set of $k$ integers is said to be a [i]complete residue system modulo[/i] $k$ if no two of its elements are congruent modulo $k$. Find all positive integers $m$ so that there are infinitely many positive integers $n$ wherein $\{ 1^n,2^n, \dots , m^n \}$ is a complete residue system modulo $m$.
Let $p=4k+3$ be a prime. Prove that if
\[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}\]
(where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$.
[i]Proposed by A. Golovanov[/i]
A $2024$-[i]table [/i]is a table with two rows and $2024$ columns containg all the numbers $1,2,\dots,4048$. Such a table is [i]evenly coloured[/i] if exactly half of the numbers in each row, and one number in each column, is coloured red. The [i]red sum[/i] in an evenly coloured $2024$-table is the sum of all the red numbers in the table.
Let $N$ be the largest number such that every $2024$-table has an even colouring with red sum $\ge N$. Determine $N$, and find the number of $2024$-tables such that every even colouring of the table has red sum $\le N$.
A sequence $x_1, x_2, ..., x_n, ...$ consists of an initial block of $p$ positive distinct integers that then repeat periodically. This means that $\{x_1, x_2, \dots, x_p\}$ are $p$ distinct positive integers and $x_{n+p}=x_n$ for every positive integer $n$. The terms of the sequence are not known and the goal is to find the period $p$. To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
(a) Knowing that $1 \le p \le 10$, find the least $n$ such that there is a strategy which allows to find $p$ revealing at most $n$ terms of the sequence.
(b) Knowing that $p$ is one of the first $k$ prime numbers, find for which values of $k$ there exist a strategy that allows to find $p$ revealing at most $5$ terms of the sequence.
If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder
$ \textbf{(A)}\ \text{never} \qquad
\textbf{(B)}\ \text{sometimes only} \qquad
\textbf{(C)}\ \text{always} \qquad$
$ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad
\textbf{(E)}\ \text{none of these}$
Show that for nonnegative integers $m$ and $n$,
$\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$
$=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta{ABC}$.
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .