Found problems: 85335
Given real numbers $a,b,c,d$ such that $\sin{a}+b >\sin{c}+d, a+\sin{b}>c+\sin{d}$, prove that $a+b>c+d$
Let a set $S$ of $n$ points be called [i]cool[/i] if:
[list]
[*] All points lie in a plane
[*] No three points are collinear
[*] There exists a triangle with three distinct vertices in $S$ such that the triangle contains another point in $S$ strictly inside it
[/list]
Define $g(S)$ for a cool set $S$ to be the sum of the number of points strictly inside each triangle with three distinct vertices in $S$. Let $f(n)$ be the minimal possible value of $g(S)$ across all cool sets of size $n$. Find
\[ f(4) + \dots + f(2020) \pmod{1000}\]
[i]Proposed by Timothy Qian[/i]
Let $ABC$ be a triangle and let $D$ be a point on $BC$ so that $AD$ bisects the angle $\angle BAC$. The common tangents of the circles $(BAD)$, $(CAD)$ meet at the point $A'$. The points $B'$, $C'$ are defined similarly. Show that $A'$, $B'$, $C'$ are collinear.
Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.
A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
What is the sum of real roots of the equation
\[
\left ( x + 1\right )\left ( x + \dfrac 14\right )\left ( x + \dfrac 12\right )\left ( x + \dfrac 34\right )= \dfrac {45}{32}?
\]
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\dfrac {3}{2}
\qquad\textbf{(D)}\ -\dfrac {5}{4}
\qquad\textbf{(E)}\ -\dfrac {7}{12}
$
Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer .
(a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$.
(b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.
What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$?
$
\textbf{(A)}\ 13
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 27
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 30
$
Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that:
[b]a)[/b] $ p_{MNPQ}\ge AC+BD. $
[b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $
[b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $
[i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
Let $ \alpha < \frac {3 \minus{} \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $ n$ and $ p > \alpha \cdot 2^n$ for which one can select $ 2 \cdot p$ pairwise distinct subsets $ S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $ \{1,2, \ldots, n\}$ such that $ S_i \cap T_j \neq \emptyset$ for all $ 1 \leq i,j \leq p$
[i]Author: Gerhard Wöginger, Austria[/i]
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
Consider a finite set of points, $\Phi$, in the $\mathbb{R}^2$, such that they follow the following properties :
[list]
[*] $\Phi$ doesn't contain the origin $\{(0,0)\}$ and not all points are collinear.
[*] If $\alpha \in \Phi$, then $-\alpha \in \Phi$, $c\alpha \notin \Phi $ for $c\neq 1$ or $-1$
[*] If $\alpha, \ \beta$ are in $\Phi$, then the reflection of $\beta$ in the line passing through the origin and perpendicular to the line containing origin and $\alpha$ is in $\Phi$
[*] If $\alpha = (a,b) , \ \beta = (c,d)$, (both $\alpha, \ \beta \in \Phi$) then $\frac{2(ac+bd)}{c^2+d^2} \in \mathbb{Z}$
[/list]
Prove that there cannot be 5 collinear points in $\Phi$
A square of side $N=n^2+1$, $n\in \mathbb{N}^*$, is partitioned in unit squares (of side $1$), along $N$ rows and $N$ columns. The $N^2$ unit squares are colored using $N$ colors, $N$ squares with each color. Prove that for any coloring there exists a row or a column containing unit squares of at least $n+1$ colors.
Let $N\ge2$ be a positive integer. Given a sequence of natural numbers $a_1,a_2,a_3,\dots,a_{N+1}$ such that for every integer $1\le i\le j\le N+1$,
$$a_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N}$$
Prove that there exist a positive integer $k\le N+1$ such that $\gcd(a_k, N) \neq 1$
There are 2010 students and 100 classrooms in the Olympiad High School. At the beginning, each of the students is in one of the classrooms. Each minute, as long as not everyone is in the same classroom, somebody walks from one classroom into a different classroom with at least as many students in it (prior to his move). This process will terminate in $M$ minutes. Determine the maximum value of $M$.
Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square?
(20 points )
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$. Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?
$\textbf{(A)} \text{ 0.21} \qquad \textbf{(B)} \text{ 0.25} \qquad \textbf{(C)} \text{ 0.29} \qquad \textbf{(D)} \text{ 0.50} \qquad \textbf{(E)} \text{ 0.79}$
For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$
(1) Find $H_1(x),\ H_2(x),\ H_3(x)$.
(2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction.
(3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$.
(4) Find $\lim_{a\to\infty} S_6(a)$.
If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.
Envelop the cube in one layer with five convex pentagons of equal areas.
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.
[i]Proposed by United Kingdom.[/i]
Given a triangle $ABC$, $BC = 2 \cdot AC$. Point $M$ is the midpoint of side $ BC$ and point $D$ lies on $AB$, with $AD = 2 \cdot BD$. Prove that the lines $AM$ and $MD$ are perpendicular.