Found problems: 85335
We call a sequence of $m$ consecutive integers a [i]friendly[/i] sequence if its first term is divisible by $1$, the second by $2$, ..., the $(m-1)^{th}$ by $m-1$, and in addition, the last term is divisible by $m^2$
Does a friendly sequence exist for (a) $m=20$ and (b) $m=11$?
On the table there are $20$ coins of weights $1,2,3,\ldots,15,37,38,39,40$ and $41$ grams. They all look alike but their colours are all distinct. Now Miss Adams knows the weight and colour of each coin, but Mr. Bean knows only the weights of the coins. There is also a balance on the table, and each comparison of weights of two groups of coins is called an operation. Miss Adams wants to tell Mr. Bean which coin is the $1$ gram coin by performing some operations. What is the minimum number of operations she needs to perform?
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]
[i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]
Urn A contains $4$ white balls and $2$ red balls. Urn B contains $3$ red balls and $3$ black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?
Let $A = {1,2,3,4,5}$ and $B = {0,1,2}$. Find the number of pairs of functions ${{f,g}}$ where both f and g map the set A into the set B and there are exactly two elements $x \in A$ where $f(x) = g(x)$. For example, the function f that maps $1 \rightarrow 0,2 \rightarrow 1,3 \rightarrow 0,4 \rightarrow 2,5 \rightarrow 1$ and the constant function g which maps each element of A to 0 form such a pair of functions.
Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$.
[i]Author: Vladimir Shmarov, Russia[/i]
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
The owner of a hardware store bought a quantity of screws in closed boxes and sells the screws separately. He never has more than one open box. At the end of the day Monday there are $2208$ screws left, at the end of the day Tuesday there are still $1616$ screws and at the end of Wednesday there are still $973$ screws. To control the employees, every night he writes down the number of screws that are in the only open box. The amount entered on Tuesday is double that of Monday. How many screws are there in each closed box if it is known that they are less than $500$?
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(f(x - y)) = f(x)f(y) + f(x) - f(y) - xy.\]
A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
[asy]
draw((0,0)--(6,0)--(6,1)--(5,1)--(5,2)--(0,2)--cycle);
draw((0,1)--(5,1));
draw((1,1)--(1,2));
draw((3,1)--(3,2));
draw((2,0)--(2,1));
draw((4,0)--(4,1));
[/asy]
$\text{(A)}\ 344 \qquad \text{(B)}\ 347 \qquad \text{(C)}\ 350 \qquad \text{(D)}\ 353 \qquad \text{(E)}\ 356$
For a function $f:[0,1]\to\mathbb R$ the secant of $f$ at points $a,b\in[0,1]$, $a<b$, is the line in $\mathbb R^2$ passing through $(a,f(a))$ and $(b,f(b))$. A function is said to intersect its secant at $a,b$ if there exists a point $c\in(a,b)$ such that $(c,f(c))$ lies on the secant of $f$ at $a,b$.
1. Find the set $\mathcal F$ of all continuous functions $f$ such that for any $a,b\in[0,1]$, $a<b$, the function $f$ intersects its secant at $a,b$.
2. Does there exist a continuous function $f\notin\mathcal F$ such that for any rational $a,b\in[0,1],a<b$, the function $f$ intersects its secant at $a,b$?
In the square $ABCD$ we consider $ E \in (AB)$, $ F \in (AD)$ and $EF \cap AC = \{P\}$. Show that:
a) $\frac{1}{AE} + \frac{1}{AF} = \frac{\sqrt2}{AP}$
b) $AP^2 \le \frac{AE \cdot AF}{2}$
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square.
Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.
(A.Myakishev, 9--10) Given triangle $ ABC$. One of its excircles is tangent to the side $ BC$ at point $ A_1$ and to the extensions of two other sides. Another excircle is tangent to side $ AC$ at point $ B_1$. Segments $ AA_1$ and $ BB_1$ meet at point $ N$. Point $ P$ is chosen on the ray $ AA_1$ so that $ AP\equal{}NA_1$. Prove that $ P$ lies on the incircle.
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 04[/b]
In an actue-angled triangle $ABC$ draws up: its circumcircle $w$
with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and
the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on
the straight lines $AQ$ and $AC$, respectively, in such a way that
the quadrilateral $AMBN$ is a parallelogram.
Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the
circumference $w_1.$
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88513]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.
Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$
The sequence $a_n$ is defined for any $n\geq 10$ by the following inductive rule:
[list]
[*] $a_{10}=5778$
[*] If $a_n=0$ then $a_{n+1}=0$.
[*] If $a_n\neq0$ then $a_{n+1}$ is the number whose base-$(n+1)$ representation equals the base $n$ representation of the number $a_n -1$.
[/list]
For example,
$a_{11}=5\cdot11^3+7\cdot11^2+7\cdot11^1+7\cdot11^0=7586$
$a_{12}=5\cdot12^3+7\cdot12^2+7\cdot12^1+6\cdot12^0=9738$
[list=a]
[*] Does there exist $n\geq10$ for which $a_n=0$?
[*] Is $a_{1,000,000}=0$?
[*] Is $a_{100^{100^{100}}}=0$?
[/list]
Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms:
$$\sum_{i<j}|x_i -x_j |.$$
Find all values of $a \in \mathbb{R}$ for which the polynomial
\begin{align*} f(x)=x^4-2x^3 + \left(5-6a^2 \right)x^2 + \left(2a^2-4 \right)x + \left(a^2 -2 \right)^2 \end{align*}
has exactly three real roots.
Which of the following could not be the unit's digit [one's digit] of the square of a whole number?
$ \text{(A)}\ 1\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 8 $
A value $x \in [0, 1]$ is selected uniformly at random. A point $(a, b)$ is called [i]friendly [/i] to $x$ if there exists a circle between the lines $y = 0$ and $y = 1$ that contains both $(a, b)$ and $(0, x)$. Find the area of the region of the plane determined by possible locations of friendly points.
If $p$ is the perimeter of an equilateral triangle inscribed in a circle, the area of the circle is:
$ \textbf{(A)}\ \frac{\pi p^2}{3} \qquad\textbf{(B)}\ \frac{\pi p^2}{9}\qquad\textbf{(C)}\ \frac{\pi p^2}{27}\qquad\textbf{(D)}\ \frac{\pi p^2}{81} \qquad\textbf{(E)}\ \frac{\pi p^2 \sqrt3}{27} $
Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.