Found problems: 85335
The pairwise greatest common divisors of five positive integers are
$$2, 3, 4, 5, 6, 7, 8, p, q, r$$
in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p + q + r$.
Triangles $ABC$ and $BDC$ are such that $\angle{ABC}=\angle{BDC}=90^{\circ}$ and $\angle{DBC}=\angle{CAB}$. Let $Q$ be a point on $\overline{BD}$ such that $\overline{QC}\perp\overline{AD}$. Suppose that $BD=15$. Then $DQ$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
How many positive integers $n$, with $n\le 2007$, yield a solution for $x$ (where $x$ is real) in the equation $\lfloor x \rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor=n$?
In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.
Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs $+$ , $-$ and $\times$. He may also use a special sign $\pm$: computing the values of the resulting expression, he chooses values $+$ or $-$ for every $\pm$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6 \}$, and $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3 \}$. Can Pete construct such an expression:
$a)$ If the numbers on the blackboard are $1, 2, 4$;
$b)$ For any collection of $100$ distinct real numbers on a blackboard?
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$.
Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.
Points $A,B,C,D,E,F$ lie in that order on semicircle centered at $O$, we assume that $AD=BE=CF$. $G$ is a common point of $BE$ and $AD$, $H$ is a common point of $BE$ and $CD$. Prove that:
\[\angle AOC=2\angle GOH.\]
We are given a 7x7 square board. In each square, one of the diagonals is traced, and then one of the two formed triangles is colored blue. What is the largest area a continuous blue component can have?
(Note: continuous blue component means a set of blue triangles connected via their edges, passing through corners is not permitted)
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.
[i]Proposed by Tonći Kokan, Croatia[/i]
Find the number of representations of the number $180$ in the form $180 =x+y+z$, where $x, y, z$ are positive integers that are proportional with some three consecutive positive integers
We say that an integer $m$ is a perfect power if there are $a\in\mathbf{Z}$, $b\in\mathbf{N}$ with $b > 1$ such that $m = a^b$.
Find all polynomials $P\in\mathbf{Z}[x]$ such that $P(n)$ is a perfect power for every $n\in\mathbf{N}$.
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
What is the smallest positive integer (in base 10) that has a digit sum of 23 in base 20, and a digit sum of 20 in base 23? (The digit sums are in base 10.)
[i]Team #8[/i]
Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0.$
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$.
a) If $\angle B=135^{\circ}$, find the length of $[BD]$.
b) Find the minimum of the length of $[BD]$.
a) Let $f_1,f_2,\ldots,f_n: \mathbb{R} \to \mathbb{R}$ be periodic functions such that the function $f: \mathbb{R} \to \mathbb{R},$ $f=f_1+f_2+\ldots+f_n$ has finite limit at $\infty.$ Prove that $f$ is constant.
b) If $a_1,a_2,a_3$ are real numbers such that $a_1 \cos(a_1x) + a_2 \cos (a_2x) + a_3 \cos(a_3x) \ge 0$ for every $x \in \mathbb{R},$ then $a_1a_2a_3=0.$
A parallelogram $ABCD$ is split by the diagonal $BD$ into two equal triangles. A regular hexagon is inscribed into the triangle $ABD$ so that two of its consecutive sides lie on $AB$ and $AD$ and one of its vertices lies on $BD$. Another regular hexagon is inscribed into the triangle $CBD{}$ so that two of its consecutive vertices lie on $CB$ and $CD$ and one of its sides lies on $BD$. Which of the hexagons is bigger?
[i]Konstantin Knop[/i]
Let $u_n$ denote the ramp function
$$ u_n (x) =\begin{cases}
-n \;\; \text{for} \;\; x \leq -n, \\
\; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\
\;n \;\; \; \text{for} \;\; n \leq x,
\end{cases}$$
and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$
Let $ABCD$ be a square piece of paper with side length $4$. Let $E$ be a point on $AB$ such that $AE=3$ and let $F$ be a point on $CD$ such that $DF=1$. Now, fold $AEFD$ over the line $EF$. Compute the area of the resulting shape.
Call a set \(\{a,b,c,d\}\) [i]epic[/i] if for any four different positive integers \(a, b, c, d\), there is a unique way to select three of them to form the sides of a triangle. Find all positive integers \(n\) such that \(\{1, 2, \ldots, 4n\}\) can be partitioned into \(n\) disjoint epic sets.
Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]
Let $a$ and $b$ be real numbers such that $$\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt2.\end{cases}$$ What is the value of $a^4 + b^4$ ?
Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!
The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.