Found problems: 85335
Let $ H $ denote the orthocenter of an acute triangle $ ABC, $ and $ A_1,A_2,A_3 $ denote the intersections of the altitudes of this triangle with its circumcircle, and $ A',B',C' $ denote the projections of the vertices of this triangle on their opposite sides.
[b]a)[/b] Prove that the sides of the triangle $ A'B'C' $ are parallel to the sides of $ A_1B_1C_1. $
[b]b)[/b] Show that $ B_1C_1\cdot\overrightarrow{HA_1} +C_1A_1\cdot\overrightarrow{HB_1} +A_1B_1\cdot\overrightarrow{HC_1} =0. $
[i]Geoghe Duță[/i]
Then number of solutions to $\cos7x=\cos5x$ in $[0,\pi]$ is________.
Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.
A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point A. At that time each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n. $
Solve the system
$$\begin{cases} x+y+z=1\\ x^5+y^5+z^5=1\end{cases}$$
in real numbers.
Let us have a convex quadrilateral $ABCD$ such that $AB=BC.$ A point $K$ lies on the diagonal $BD,$ and $\angle AKB+\angle BKC=\angle A + \angle C.$ Prove that $AK \cdot CD = KC \cdot AD.$
Andrej and Barbara play the following game with two strips of newspaper of length $a$ and $b$. They alternately cut from any end of any of the strips a piece of length $d$. The player who cannot cut such a piece loses the game. Andrej allows Barbara to start the game. Find out how the lengths of the strips determine the winner.
We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)
The real positive numbers $a_1, a_2,a_3$ are three consecutive terms of an arithmetic progression, and similarly, $b_1, b_2, b_3$ are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths $a_1, a_2, a_3$ as bases, and other three segments of lengths $b_1, b_2, b_3$ as altitudes, to construct three rectangles of equal area ?
Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.
A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cyclist $15$ minutes after meeting the bus and another $15$ minutes later caught up with the pedestrian. At what time did the cyclist and the pedestrian meet? (The speeds and directions of movement of all participants were equal, the pedestrian and the motorcyclist were moving in the direction of city $A$.)
Is there a set of distinct integers $X$ containing all the primes less than $2007$ such that the product of the elements of $X$ equals the sum of the squares of those elements?
Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.
Let $ABC$ be a triangle with $\angle B=90$. The incircle of $ABC$ touches the side $BC$ at $D$. The incenters of triangles $ABD$ and $ADC$ are $X$ and $Z$ , respectively. The lines $XZ$ and $AD$ are intersecting at the point $K$. $XZ$ and circumcircle of $ABC$ are intersecting at $U$ and $V$. Let $M$ be the midpoint of line segment $[UV]$ . $AD$ intersects the circumcircle of $ABC$ at $Y$ other than $A$. Prove that $|CY|=2|MK|$ .
Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.
Find all pairs of real numbers $(x,y)$ satisfying:
\[x+y^2=y^3,\]\[y+x^2=x^3.\]
Determine all points on the straight line which joins $(4, 11)$ to $(16, 1)$ and whose coordinates are positive integers.
From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all
original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the
cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original
numbers).
a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points)
b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points)
[i](Alexey Tolpygo)[/i]
Polygon $A_0A_1...A_{n-1}$ satisfies the following:
$\bullet$ $A_0A_1 \le A_1A_2 \le ...\le A_{n-1}A_0$ and
$\bullet$ $\angle A_0A_1A_2 = \angle A_1A_2A_3 = ... = \angle A_{n-2}A_{n-1}A_0$ (all angles are internal angles).
Prove that this polygon is regular.
Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers).
Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.
Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)
In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?
Let $ABC$ be an acute triangle with orthocenter $H$. Points $A_1$, $B_1$, $C_1$ are chosen in the interiors of sides $BC$, $CA$, $AB$, respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$. Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$, $B_2 = \overline{BH} \cap \overline{C_1A_1}$, and $C_2 = \overline{CH} \cap \overline{A_1B_1}$.
Prove that triangle $A_2B_2C_2$ has orthocenter $H$.
[i]Ankan Bhattacharya[/i]
Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices.
(B.Frenkin)
Consider a quadrilateral with $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$ and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$.