Found problems: 85335
Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that
$$f(a) \equiv g(a+m_p) \pmod p$$
holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that
$$f(x)=g(x+r).$$
For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.
Three lines are drawn in a plane.
Which of the following could NOT be the total number of points of intersections?
(A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) They all could.
Let $HOPKINS$ be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than $180^{\circ}$) with area $1876$ such that all of its side lengths are greater than $5$, $OP=20$, and $KI=22$. Arcs with radius $2$ are drawn inside $HOPKINS$ with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside $HOPKINS$ but outside the sectors.
Find all positive integers $(m,n)$ such that
\[n^{n^{n}}=m^{m}.\]
Find all polynomials $P(x)$ with real coefficients such that
\[(x-2010)P(x+67)=xP(x) \]
for every integer $x$.
Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $
Let $ABCD$ be a parallelogram, $E$ a point on the line $AB$, beyond $B, F$ a point on the line $AD$, beyond $D$, and $K$ the point of intersection of the lines $ED$ and $BF$. Prove that quadrilaterals $ABKD$ and $CEKF$ have the same area.
Let $\triangle ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC,$ and $CA$ respectively, such that:
\[
\frac{AP}{AB} = \frac{BQ}{BC} = \frac{CR}{CA} = \frac{1}{n}
\]
for $n \in \mathbb{N}$. The segments $AQ$ and $CP$ intersect at $D$, the segments $BR$ and $AQ$ intersect at $E$, and the segments $BR$ and $CP$ intersect at $F$. Compute the ratio:
\[
\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}.
\]
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
In a acutangle triangle $ABC, \angle B>\angle C$. Let $D$ the foot of the altitude from $A$ to $BC$ and $E$ the foot of the perpendicular from $D$ to $AC$. Let $F$ a point in $DE$. Prove that $AF$ and $BF$ are perpendiculars if and only if $EF\cdot DC=BD\cdot DE$.
The numerical value of the following integral $$\int^1_0 (-x^2 + x)^{2017} \lfloor 2017x \rfloor dx$$ can be expressed in the form $a\frac{m!^2}{ n!}$ where a is minimized. Find $a + m + n$.
(Note $\lfloor x\rfloor$ is the largest integer less than or equal to x.)
$20$ football teams participate in the championship. What minimal number of the games should be played to provide the property:
[i] from the three arbitrary teams we can find at least on pair that have already met in the championship.[/i]
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
Let $ab+bc+ca=1$. Show that
\[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\]
Prove that if $ n\geq 4$, $ n\in\mathbb Z$ and $ \left \lfloor \frac {2^n}{n} \right\rfloor$ is a power of 2, then $ n$ is also a power of 2.
Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear.
(Malik Talbi)
A sequence $a_1,\ldots,a_n$ of positive integers is given. For each $l$ from $1$ to $n-1$ the array $(gcd(a_1,a_{1+l}),\ldots,gcd(a_n,a_{n+l}))$ is considered, where indices are taken modulo $n$. It turned out that all this arrays consist of the same $n$ pairwise distinct numbers and differ only,possibly, by their order.
Can $n$ be a) $21$ b) $2021$
Let $ABC$ be a triangle with $AB = 7, BC = 5,$ and $CA = 6$. Let $D$ be a variable point on segment $BC$, and let the perpendicular bisector of $AD$ meet segments $AC, AB$ at $E, F,$ respectively. It is given that there is a point $P$ inside $\triangle ABC$ such that $\frac{AP}{PC} = \frac{AE}{EC}$ and $\frac{AP}{PB} = \frac{AF}{FB}$. The length of the path traced by $P$ as $D$ varies along segment $BC$ can be expressed as $\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right)$, where $m$ and $n$ are relatively prime positive integers, and angles are measured in radians. Compute $100m + n$.
[i]Proposed by Edward Wan[/i]
Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.
The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at $D,E,F$, respectively. Let $G$ be a point on the incircle such that $FG$ is a diameter. The lines $EG$ and $FD$ intersect at $H$. Prove that $CH\parallel AB$.
The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$ and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 5)[/i]
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$
You have $9$ colors of socks and $5$ socks of each type of color. Pick two socks randomly. What is the probability that they are the same color?
Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$?
$\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111$