Found problems: 85335
Find all polynomials $P$ with real coefficients satisfying that there exist infinitely many pairs $(m, n)$ of coprime positives integer such that $P(\frac{m}{n})=\frac{1}{n}$.
[i]
Proposed by usjl[/i]
If $a$, $b$, $c > 0$, prove that$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}$$
2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative.
(Authors: N. Agahanov & I. Bogdanov)
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
[i]Proposed by David Altizio[/i]
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$.
Prove that $m$ is a divisor of $n$.
Real numbers $x$, $y$, and $z$ satisfy the inequalities
$$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$
Which of the following numbers is nessecarily positive?
$\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\
\textbf{(E) } y+z$
Let $a, b, c, d$ be non-negative real numbers such that \[\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.\]
Prove that \[3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.\][i]Vasile Cîrtoaje and Leonard Giugiuc[/i]
An acute-angled triangle $AKL$ is given on a plane. Consider all rectangles $ABCD$ circumscribed to triangle $AKL$ such that point $K$ lies on side $BC$ and point $L$ lieson side $CD$. Find the locus of the intersection $S$ of the diagonals $AC$ and $BD$.
One quadrant of the Cartesian coordinate system is tiled with $1\times 2$ dominoes. The dominoes don’t overlap with each other, they cover the entire quadrant and they all fit in the quadrant. Farringdon the flea is initially sitting at the origin and is allowed to jump from one corner of a domino to the opposite corner any number of times. Is it possible that the dominoes are arranged in such a way that Farringdon is unable to move to a distance greater than 2023 from the origin?
Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$
ABCD=cyclic quadrilateral,$AC\cap BD=X$
AA'$\perp $BD,A'$\in$BD
CC'$\perp $BD,C'$\in$BD
BB'$\perp $AC,B'$\in$AC
DD'$\perp $AC,D'$\in$AC
Prove that:
a)Prove that perpendiculars from midpoints of the sides to the opposite sides are concurrent.The point is called Mathot Point
b)A',B',C',D' are concyclic
c)If O'=circumcenter of (A'B'C') prove that O'=midpoint of the line that connects the orthocente of triangle XAB and XCD
d)O' is the Mathot Point
Let $ABC$ be an isosceles triangle with $AB = AC$. Let P be a point in the interior of $ABC$ such that $PB > PC$ and $\angle PBA = \angle PCB$. Let $M$ be the midpoint of the side $BC$. Let $O$ be the circumcenter of the triangle $APM$. Prove that $\angle OAC=2 \angle BPM$ .
Bread draws a circle. He then selects four random distinct points on the circumference of the circle to form a convex quadrilateral. Kwu comes by and randomly chooses another 3 distinct points (none of which are the same as Bread's four points) on the circle to form a triangle. Find the probability that Kwu's triangle does not intersect Bread's quadrilateral, where two polygons intersect if they have at least one pair of sides intersecting.
[i]Proposed by Nathan Cho[/i]
What's the smallest positive integer \( n > 3 \), for which there does [b]not[/b] exist a (not necessarily convex) \( n \)-gon such that all its diagonals have equal lengths? A diagonal of any polygon is defined as a segment connecting any two non-adjacent vertices of the polygon.
[i]Proposed by Anton Trygub[/i]
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$
Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$
Let $n>100$ be an integer. The numbers $1,2 \ldots, 4n$ are split into $n$ groups of $4$. Prove that there are at least $\frac{(n-6)^2}{2}$ quadruples $(a, b, c, d)$ such that they are all in different groups, $a<b<c<d$ and $c-b \leq |ad-bc|\leq d-a$.
There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.
Let $p \geqslant 5$ be a prime and $S = \left\{ 1, 2, \ldots, p \right\}$. Define $r(x,y)$ as follows: \[ r(x,y) = \begin{cases} y - x & y \geqslant x \\ y - x + p & y < x \end{cases}.\]
For a nonempty proper subset $A$ of $S$, let $$f(A) = \sum_{x \in A} \sum_{y \in A} \left( r(x,y) \right)^2.$$A [i]good[/i] subset of $S$ is a nonempty proper subset $A$ satisfying that for all subsets $B \subseteq S$ of the same size as $A$, $f(B) \geqslant f(A)$. Find the largest integer $L$ such that there exists distinct good subsets $A_1 \subseteq A_2 \subseteq \ldots \subseteq A_L$.
[i]Proposed by Bin Wang[/i]
The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
In the rectangle $ABCD$, the ratio of the lengths of sides $BC$ and $AB$ is equal to $\sqrt{2}$. Point $X$ is marked inside this rectangle so that $AB=BX=XD$. Determine the measure of angle $BXD$.
Given a semicirle with center $O$ an arbitrary inner point of the diameter divides it into two segments. Let there be semicircles above the two segments as visible in the below figure. The line $\ell$ passing through the point $A$ intersects the semicircles in $4$ points: $B, C, D$ and $E$. Show that the segments $BC$ and $DE$ have the same length.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/86a369d54fef7e25a51fea6481c0b5e7dd45ff.png[/img]
Two tangents $CA$ and $CB$ are drawn to a circle ($A$ and $B$ being the tangent points). Consider a “triangle” bounded by an arc $AB$ (the smaller one) and segments $CA$ and $CB$. Prove that the length of any segment inside the triangle is not greater than the length of $CA = CB$.
(Folklore)
Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]
Given a prime $ p = 8k+1 $ for some integer $ k $. Let $ r $ be the remainder when $ \binom{4k}{k} $ is divided by $ p $. Prove that $ \sqrt{r} $ is not an integer.
[i]Proposed by Evan Chen[/i]