Found problems: 85335
Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$, such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$. Let $P$ be a point on the line $AC$. Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$.
Triangle $ABC$ has side lengths $13$, $14$, and $15$. Let $E$ be the ellipse that encloses the smallest area which passes through $A, B$, and $C$. The area of $E$ is of the form $\frac{a \sqrt{b}\pi}{c}$ , where $a$ and $c$ are coprime and $b$ has no square factors. Find $a + b + c$.
There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$. Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have :
$$f(an+b)=g(cn+d)$$
Prove that at least one of the followings hold.
$i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$
$ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$
(Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$)
Proposed by [i]Navid Safaii[/i]
Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$
In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.
Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
In a $100\times 100$ table $110$ unit squares are marked. Is it always possible to rearrange rows and columns so that all the marked unit squares
are above the main diagonal or on it?
Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs.
a) Determine the maximum possible area of a trio.
b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon.
c) Determine the total number of triangles associated with the hexagon.
Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that
\[\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.\]
A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.
[i]Proposed by Evan Chen[/i]
Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?
Let $ A_1A_2$ be the external tangent line to the nonintersecting cirlces $ \omega_1(O_1)$ and $ \omega_2(O_2)$,$ A_1\in\omega_1$,$ A_2\in\omega_2$.Points $ K$ is the midpoint of $ A_1A_2$.And $ KB_1$ and $ KB_2$ are tangent lines to $ \omega_1$ and $ \omega_2$,respectvely($ B_1\neq A_1$,$ B_2\neq A_2$).Lines $ A_1B_1$ and $ A_2B_2$ meet in point $ L$,and lines $ KL$ and $ O_1O_2$ meet in point $ P$.
Prove that points $ B_1,B_2,P$ and $ L$ are concyclic.
Let $L$ be the midpoint of the minor arc $AC$ of the circumcircle of an acute-angled triangle $ABC$. A point $P$ is the projection of $B$ to the tangent at $L$ to the circumcircle. Prove that $P$, $L$, and the midpoints of sides $AB$, $BC$ are concyclic.
Let $A = \{1,2,\ldots,2011\}$. Find the number of functions $f$ from $A$ to $A$ that satisfy $f(n) \le n$ for all $n$ in $A$ and attain exactly $2010$ distinct values.
If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that:
$ \textbf{(A)}\ \text{they form an arithmetic progression} \\
\textbf{(B)}\ \text{they form a geometric progression} \\
\textbf{(C)}\ \text{they are unequal} \\
\textbf{(D)}\ \text{they are all negative numbers} \\
\textbf{(E)}\ \text{only b is negative and a and c are positive}$
For any $p, q \in N$, we can express $\frac{p}{q}$ as the base $10$ decimal $x_1x_2... x_{\ell}.x_{\ell+1}... x_a \overline{y_1y_2... y_b}$, with the digits $y_1, . . . y_b$ repeating. In other words, $\frac{p}{q}$ can be expressed with integer part $x_1x_2... x_{\ell}$ and decimal part $0.x_{\ell+1}... x_a \overline{y_1y_2... y_b}$. Given that $\frac{p}{q}= \frac{(2021)^{2021}}{2021!}$ , estimate the minimum value of $a$. If $E$ is the exact answer to this question and $A$ is your answer, your score is given by $\max \, \left(0, \left\lfloor 25 - \frac{1}{10}|E - A|\right\rfloor \right)$.
For every $n\geq 1$ define $x_n$ by $$x_n=\int_0^1 \ln(1+x+x^2+\dots +x^n)\cdot\ln\frac{1}{1-x}\mathrm dx.$$
a) Show that $x_n$ is finite for every $n\geq 1$ and $\lim_{n\rightarrow\infty}x_n=2$.
b) Calculate $\lim_{n\rightarrow\infty}\frac{n}{\ln n}(2-x_n)$.
Construct a triangle $ABC$, given the length $c$ of its side $AB$, the radius $r$ of its inscribed circle, and the radius $r_c$ of its ex-circle tangent to the side $AB$ and the extensions of $BC$ and $CA$.
For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g $P(6)=36$). For every positive integer $n$ define the sequence
$$a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n))\quad (k=1,2,\dots,2016)$$
Determine whether for every set $S\subset\{1,2,\dots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:
For every $k$ with $1\leq k\leq 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.
Given six points $ A, B, C, D, E, F $ such that $ \triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA $ and let $ I $ be the incenter of $ \triangle ABC. $ Prove that the circumcenter of $ \triangle AID, \triangle BIE, \triangle CIF $ are collinear.
[i]Proposed by Telv Cohl[/i]
On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$-th step ($i \ge 2$) we merge some three heaps containing more than $i$ stones in total and remove $i$ stones from the new heap. Assume that after a number of steps a single heap of $p$ stones remains on the table. Show that the number $p$ is a perfect square if and only if so are both $2k + 2$ and $3k + 1$. Find the least $k$ with this property.
Find all positive integers $n$ that are quadratic residues modulo all primes greater than $n$.
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ?
[asy]
draw((0,2)--(2,2)--(2,0)--(0,0)--cycle);
draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle);
label("$a$",(-0.1,0.15));
label("$b$",(-0.1,1.15));
[/asy]
$\textbf{(A)}\hspace{.05in}\dfrac15 \qquad \textbf{(B)}\hspace{.05in}\dfrac25 \qquad \textbf{(C)}\hspace{.05in}\dfrac12 \qquad \textbf{(D)}\hspace{.05in}1 \qquad \textbf{(E)}\hspace{.05in}4 $
Let $n$ be a positive integer. David has six $n\times n$ chessboards which he arranges in an $n\times n\times n$ cube. Two cells are "aligned" if they can be connected by a path of cells $a=c_1,\ c_2,\ \dots,\ c_m=b$ such that all consecutive cells in the path share a side, and the sides that the cell $c_i$ shares with its neighbors are on opposite sides of the square for $i=2,\ 3,\ \dots\ m-1$.
Two towers attack each other if the cells they occupy are aligned. What is the maximum amount of towers he can place on the board such that no two towers attack each other?
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .