Found problems: 85335
Let $ABC$ be a triangle with sides $BC = 13$, $CA = 14$ and $AB = 15$. We denote by $I$ the intersection point of the angle bisectors and $M$ to the midpoint of $AB$. The line $IM$ cuts at $P$ at the altitude drawn from $C$. Find the length of $CP$.
For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$, $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$. Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$.
How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?
$\textbf{(A) } 96 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 98 \qquad \textbf{(D) } 102 \qquad \textbf{(E) } 120 $
Let $a_0, b_0, c_0$ be complex numbers, and define \begin{align*}a_{n+1} &= a_n^2 + 2b_nc_n \\ b_{n+1} &= b_n^2 + 2c_na_n \\ c_{n+1} &= c_n^2 + 2a_nb_n\end{align*}for all nonnegative integers $n.$
Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that $$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$
In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.
Given a real number $a$, the [i]floor[/i] of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is
\[
x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \, ?
\]
We color all points of a plane using $3$ colors. Prove that there are at least two points of the plane having same colours and with distance among them $1$.
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
If $ a^x \equal{} c^q \equal{} b$ and $ c^y \equal{} a^z \equal{} d$, then
$ \textbf{(A)}\ xy \equal{} qz \qquad\textbf{(B)}\ \frac {x}{y} \equal{} \frac {q}{z} \qquad\textbf{(C)}\ x \plus{} y \equal{} q \plus{} z \qquad\textbf{(D)}\ x \minus{} y \equal{} q \minus{} z$
$ \textbf{(E)}\ x^y \equal{} q^z$
The positive numbers $a,b,c,A,B,C$ satisfy a condition $$a + A = b + B = c + C = k$$ Prove that $$aB + bC + cA \le k^2$$
A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is \( b \) and \( a \) is entered, the new password is \( c \), which is determined by \( b \) and \( a \) (naturally, the person does not know \( c \) or \( b \)). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.
Proposed by Reza Tahernejad Karizi
Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$
[i]Proposed by Daniel Liu[/i]
Let $\sigma(\cdot)$ denote the divisor sum function and $d(\cdot)$ denote the divisor counting function. Find all positve integers $n$ such that $\sigma(d(n))=n.$
[i]Andrei Bâra[/i]
Let $ABC$ be an acute-angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let $N$ be the foot of the perpendicular from $A$ to $BC$. If $NP = NQ$ then prove that $N$ is also the centre of the circumcircle of $APQ$.
Determine the smallest positive integer $n$ that is equal to the sum of $11$ consecutive positive integers, the sum of $12$ consecutive positive integers and the sum of $13$ consecutive positive integers.
A convex hexagon $ABCDEF$ is such that
$$AB=BC \quad CD=DE \quad EF=FA$$
and
$$\angle ABC=2\angle AEC \quad \angle CDE=2\angle CAE \quad \angle EFA=2\angle ACE$$
Show that $AD$, $CF$ and $EB$ are concurrent.
It is given that $r=\left(3\left(\sqrt6-1\right)-4\left(\sqrt3+1\right)+5\sqrt2\right)R$ where $r$ and $R$ are the radii of the inscribed and circumscribed spheres in a regular $n$-angled pyramid. If it is known that the centers of the spheres given coincide,
(a) find $n$;
(b) if $n=3$ and the lengths of all edges are equal to a find the volumes of the parts from the pyramid after drawing a plane $\mu$, which intersects two of the edges passing through point $A$ respectively in the points $E$ and $F$ in such a way that $|AE|=p$ and $|AF|=q$ $(p<a,q<a)$, intersects the extension of the third edge behind opposite of the vertex $A$ wall in the point $G$ in such a way that $|AG|=t$ $(t>a)$.
If $a,b,c$ are positive reals such that $a+b+c=1$, prove that \[ 10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\geq 1 . \]
Let $ 0 $ be a root for a polynom $ f\in\mathbb{R}[X] $ that has the property that $ f(X^2-X+1) =f^2(X)-f(X)+1. $
Determine this polynom.
[i]Nelu Chichirim[/i]
In the tetrahedron $ABCD$, the point $E$ is the projection of the point $D$ on the plane $(ABC)$. Prove that the following statements are equivalent:
a) $C = E$ or $CE \parallel AB$
b) For each point M belonging to the segment $CD$, the following equation is satisfied
$$S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}$$
where $S_{\vartriangle XYZ}$ means the area of triangle $XYZ$.
Triangle $\vartriangle ABC$ is inscribed in circle $O$ and has sides $AB = 47$, $BC = 69$, and $CA = 34$. Let $E$ be the point on $O$ such that $\overline{AE}$ and $\overline{BC}$ intersect inside $O$, $8$ units away from $B$. Let $P$ and $Q$ be the points on $\overleftrightarrow{BE}$ and $\overleftrightarrow{CE}$, respectively, such that $\angle EPA$ and $\angle EQA$ are right angles. Suppose lines $\overleftrightarrow{AP}$ and $\overleftrightarrow{AQ}$ respectively intersect $O$ again at $X$ and $Y$ . Compute the distance $XY$.
Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system
$$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\
a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\
a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$
has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.
Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively.
Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.
Let $ABC$ be a triangle with $AB=3$ and $AC=4$. It is given that there does not exist a point $D$, different from $A$ and not lying on line $BC$, such that the Euler line of $ABC$ coincides with the Euler line of $DBC$. The square of the product of all possible lengths of $BC$ can be expressed in the form $m+n\sqrt p$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $100m+10n+p$.
Note: For this problem, consider every line passing through the center of an equilateral triangle to be an Euler line of the equilateral triangle. Hence, if $D$ is chosen such that $DBC$ is an equilateral triangle and the Euler line of $ABC$ passes through the center of $DBC$, then consider the Euler line of $ABC$ to coincide with "the" Euler line of $DBC$.
[i]Proposed by Michael Ren[/i]