Found problems: 85335
Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.
Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$
In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each.
Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$?
If so, determine all values of $a$ for which this is possible.
In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.
For every real number $x$, let $[x]$ be the greatest integer less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always
$\textbf{(A) }6W\qquad\textbf{(B) }6[W]\qquad\textbf{(C) }6([W]-1)\qquad\textbf{(D) }6([W]+1)\qquad \textbf{(E) }-6[-W]$
Points $E$ and $F$ lie inside a square $ABCD$ such that the two triangles $ABF$ and $BCE$ are equilateral. Show that $DEF$ is an equilateral triangle.
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:
$$q |(x+1)^p-x^p$$
If and only if $$q \equiv 1 \pmod p$$
Evaluate the integral $\int_0^{\infty} \frac{\log(x+2)}{x^2+3x+2}\mathrm{d}x$.
Let $X$ be the set of consisting of twenty positive integers $n,n+2,...,n+38$. The smallest value of $n$ for which any three numbers $a,b,c \in X$, not necessarily distinct, form the sides of an acute-angled triangle is:
On the side $DE$ and on the diagonal $BE$ of the regular pentagon $ABCDE$ we consider the squares $DEFG$ and $BEHI$.
[list=a]
[*] Prove that $A,I,$ and $G$ are collinear.
[*] Prove that on this line lies also the centre $O$ of the square $BDJK$.
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Let $ABC$ be a triangle, $H$ the orthocenter and $M$ the midpoint of $AC$. Let $\ell$ be the parallel through $M$ to the bisector of $\angle AHC$. Prove that $\ell$ divides the triangle in two parts of equal perimeters.
[i]Pedro Marrone, Panamá[/i]
Given is a triangle $ABC$ with altitude $AH$, diameter of the circumcircle $AD$ and incenter $I$. Prove that $\angle BIH = \angle DIC$.
A cube is formed from $n^3$ ($n \ge 2$) unit cubes, each painted white on five randomly selected sides. This cube is dipped into paint remover and broken into the original unit cubes. What is the expected number of these unit cubes with exactly four sides painted white?
[b]Problem Section #1
b) Let $a, b$ be positive integers such that $b^n +n$ is a multiple of $a^n + n$ for all positive integers $n$. Prove that $a = b.$
Let $X=\mathbb{Z}^3$ denote the set of all triples $(a,b,c)$ of integers. Define $f: X \to X$ by \[ f(a,b,c) = (a+b+c, ab+bc+ca, abc) . \]
Find all triples $(a,b,c)$ such that \[ f(f(a,b,c)) = (a,b,c) . \]
Let $MN$ be a chord of the circle $\Gamma$ and let $S$ be the midpoint of $MN$. Let $A, B, C, D$ be
points on $\Gamma$ such that $AC$ and $BD$ intersect at $S$ and $A$ and $B$ are on the same side of $MN$.
Let $d_A, d_B, d_C , d_D$ be the distances from $MN$ to $A, B, C,$ and $D,$ respectively. Prove that $\frac{1}{d_A}+\frac{1}{d_D}=\frac{1}{d_B}+\frac{1}{d_C}$.
The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can:
a) To be equal to only one meter.
b) Be equal to the length of the diagonal $ AC $.
Find all triples of positive integers $x, y, z$ satisfying
\[\frac{1}{x} +\frac{1}{y} + \frac{1}{z} = \frac{4}{5} .\]
Over all natural numbers $n$ with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of $s(n)/n$, where $s(n)$ denotes the sum of the divisors of $n$. What is the value of $d(d(n))$, where $d(n)$ is the the number of divisors of $n$?
Show that there is a natural number $ n $ that satisfies the following inequalities:
$$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$
Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a [i]rook set[/i] as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.\\
\\ By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.
We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$
($a_n b_m \neq 0$) similar if the following conditions hold:
$(i)$ $n = m$;
$(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$.
Let $P(x)$ and $Q(x)$ be similar polynomials with integer coefficients. Given that
$P(16) = 3^{2012}$, find the smallest possible value of $|Q(3^{2012})|$.
[i]Proposed by Milos Milosavljevic[/i]
In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$
Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$.
[i]Proposed by Gogi Khimshiashvili, Georgia[/i]