Found problems: 85335
In the triangle $ABC$, for which $AC <AB <BC$, on the sides $AB$ and $BC$ the points $K$ and $N$ were chosen, respectively, that $KA = AC = CN$. The lines $AN$ and $CK$ intersect at the point $O$. From the point $O$ held the segment $OM \perp AC $ ($M \in AC$) . Prove that the circles inscribed in triangles $ABM$ and $CBM$ are tangent.
(Igor Nagel)
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.
A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.
The area of a convex quadrilateral $ABCD$ is $18$. If $|AB|+|BD|+|DC|=12$, then what is $|AC|$?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 6\sqrt 3
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 6\sqrt 2
$
Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.
A board is called ''Guapo'' if it can be totally covered by pieces equal to that shown in the picture, without gaps and without overlaps or pieces that cover areas outside the board. Is possible to rotate or reflect the pieces.
[img]https://imgur.com/o6jX1JO.jpeg[/img]
Find all positive integers $n$ such that a board $n \times (n+1)$ is ''Guapo''.
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.
In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$.
Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.
In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.
Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.
Inside a triangle $ABC$ three circles with radius $1$ are drawn. (Circles can be tangent to each other and to the sides of the triangle, but can not have any common internal points.)
Find the biggest value of $r$ for which one can state that he can always draw a fourth circle inside the triangle of radius $r$, which does not intersect three already drawn circles.
In a school there are $n$ classes and $k$ student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than $k$ students. If $k-1$ is not a perfect square, prove that there exist a student that has attended in at least $\sqrt k$ classes.
[i]Proposed by Mohammad Moshtaghi Far, Kian Shamsaie[/i] [b]Rated 4[/b]
Let $ \mathcal{F}$ be a nonempty family of sets with the following properties:
(a) If $ X \in \mathcal{F}$, then there are some $ Y \in \mathcal{F}$ and $ Z \in \mathcal{F}$ such that $ Y \cap Z =\emptyset$ and $ Y \cup Z=X$.
(b) If $ X \in \mathcal{F}$, and $ Y \cup Z =X , Y \cap Z=\emptyset$, then either $ Y \in \mathcal{F}$ or $ Z \in \mathcal{F}$.
Show that there is a decreasing sequence $ X_0 \supseteq X_1 \supseteq X_2 \supseteq ...$ of sets $ X_n \in \mathcal{F}$ such that \[ \bigcap_{n=0}^{\infty} X_n= \emptyset.\]
[i]F. Galvin[/i]
In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$
$\textbf{(A) } 76 \qquad
\textbf{(B) } 77 \qquad
\textbf{(C) } 78 \qquad
\textbf{(D) } 79 \qquad
\textbf{(E) } 80 $
In the equilateral triangle $ABC, M, N$ are the midpoints of the sides $AB, AC$, respectively. The line $MN$ intersects the circumcircle of $\vartriangle ABC$ at $K$ and $L$ and the lines $CK$ and $CL$ meet the line $AB$ at $P$ and $Q$, respectively. Prove that $PA^2 \cdot QB = QA^2 \cdot PB$.
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
Find the number of distinct permutations of $ITEST$.
[i](.1 point)[/i]
Determine all positive integers $n$ for which there exists a set $S$ with the following properties:
(i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$;
(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that
\[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \]
b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$.
[i]Mihai Baluna[/i]
Solve the system of equations
\[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]
Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.
$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$.
[hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]
Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that \[\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.\]