Found problems: 85335
A number of $n > m \geq 1$ soccer teams play a full tournament, each team meeting (once) each other. Points are awarded: $2$ for a victory, $1$ for a tie and $0$ for a loss. At the end, each team has won half of its points against the $m$ teams placed last (including each of these teams, who won half of its points against the other $m-1$).
Find all possible values for $n$ and $m$, supported with examples of such tournaments.
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$.
Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible.
Justify your answer
The fractions $ \frac{ax\plus{}b}{cx\plus{}d}$ and $ \frac{b}{d}$ are unequal if:
$ \textbf{(A)}\ a\equal{}c\equal{}1, x\neq 0 \qquad
\textbf{(B)}\ a\equal{}b\equal{}0 \qquad
\textbf{(C)}\ a\equal{}c\equal{}0 \\
\textbf{(D)}\ x\equal{}0 \qquad
\textbf{(E)}\ ad\equal{}bc$
Find all real numbers $x$ such that exactly one of the four numbers $x-\sqrt 2$, $x-\dfrac{1}{x}$, $x+\dfrac{1}{x}$ and $x^2+2\sqrt{2}$ is [b]not[/b] an integer.
Let $S, T$ be the circumcenter and centroid of triangle $ABC$, respectively. $M$ is a point in the plane of triangle $ABC$ such that $90^\circ \leq \angle SMT < 180^\circ$. $A_1, B_1, C_1$ are the intersections of $AM, BM, CM$ with the circumcircle of triangle $ABC$ respectively. Prove that $MA_1 + MB_1 + MC_1 \geq MA + MB + MC.$
A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:
$ \textbf{(A)}\ \frac{2 \plus{} \sqrt{2}}{3} \qquad
\textbf{(B)}\ \frac{2 \minus{} \sqrt{2}}{2}\qquad
\textbf{(C)}\ \frac{1 \plus{} \sqrt{2}}{2}\qquad
\textbf{(D)}\ \frac{1 \plus{} \sqrt{2}}{3}\qquad
\textbf{(E)}\ \frac{2 \minus{} \sqrt{2}}{3}$
Let $n$ be a natural number. Prove that the number of all non-isosceles triangles with lengths of their sides equal to natural numbers and a perimeter $2n$ is
$[\frac{n^2-6n+12}{12}]$.
A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.
The sum of $10$ positive integer numbers is equal to $300$. The product of their factorials is a perfect tenth power of some positive integer. Prove that all $10$ numbers are equal to each other.
[i]Proposed by Pavlo Protsenko[/i]
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
Let $I, O, \omega, \Omega$ be the incenter, circumcenter, the incircle, and the circumcircle, respectively, of a scalene triangle $ABC$. The incircle $\omega$ is tangent to side $BC$ at point $D$. Let $S$ be the point on the circumcircle $\Omega$ such that $AS, OI, BC$ are concurrent. Let $H$ be the orthocenter of triangle $BIC$. Point $T$ lies on $\Omega$ such that $\angle ATI$ is a right angle. Prove that the points $D, T, H, S$ are concyclic.
[i]Proposed by ltf0501[/i]
An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.
In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.
Find all natural numbers $n$ for which the sum of digits of $5^n$ equals $2^n$.
Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.
Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.
[i]Bogdan Enescu, Romania[/i]
Let $a_1, a_2, a_3$ be three distinct real numbers. Define
$$\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\
b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\
b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}$$
Prove that $$1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)$$
When does equality hold?
Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$.
Đỗ Thanh Sơn
PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.
The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this
square trinomial has?
The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$.
(a) How many pages could the notebook originally have been?
(b) What page numbers can be on the torn sheet?
(Walther Janous)
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
On the surface of the planet lives one inhabitant, that can move with the speed not greater than $u$. A spaceship approaches to the planet with its speed $v$. Prove that if $v/u > 10$ , the spaceship can find the inhabitant, even it is trying to hide.
A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$
Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.
In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?
Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.