Found problems: 85335
Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$
Consider $\left(\triangle_n=A_nB_nC_n\right)_{n\ge 1}$. We define points $A_n',B_n',C_n'$ in sides $C_nB_n,A_nC_n,B_nA_n$ such that
$$(n+1)B_nA_n'=C_nA_n',~(n+1)C_nB_n'=A_nB_n',~(n+1)A_nC_n'=B_nC_n'$$$\triangle_{n+1}$ is defined by the intersections of $A_nA_n',B_nB_n',C_nC_n'$. If $S_n$ denotes the area of $\triangle_n$. Find $\frac{S_1}{S_{2020}}$.
[i]Proposed by Alejandro Madrid, Valle[/i]
Let $\omega=\cos\frac{\pi}{5}+\text{i}\sin\frac{\pi}{5}$, which equation has roots $\omega,\omega^3,\omega^7,\omega^9$?
$\text{(A)}x^4+x^3+x^2+x+1=0\qquad\text{(B)}x^4-x^3+x^2-x+1=0$
$\text{(C)}x^4-x^3-x^2+x+1=0\qquad\text{(D)}x^4+x^3+x^2-x+1=0$
Find positive real numbers $x,y,z$ that are solutions of the system
$x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.
Prove that $\sqrt 2$ is irrational.
Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a
$\frac a2x^2+bx+c=0$ between $r$ and $s$.
Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$.
The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$.
The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$.
The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$.
(Yes, these definitions have the symmetries you would expect.)
Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.
Positive integers $k$ and $n$ are given such that $3 \le k \le n$.Prove that among any $n$ pairwise distinct real numbers one can choose either $k$ numbers with positive sum, or $k-1$ numbers with negative sum.
[i]Proposed by Mykhailo Shtandenko[/i]
Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$.
Let $T$ be the area of $\triangle DEF$. Prove the following inequality.
$$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$
Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.
Given a square $5$ x $7$ board and $35$ pieces, each piece is formed by $3$ squares like below:
[size=75][center][img]https://i.ibb.co/hFDhp9p/Screenshot-2023-03-26-061057.png[/img][/center][/size]
Can we fill the board with $35$ pieces such that there are exactly $3$ pieces superimpose on every square of the given board?
[i][color=#E06666]Note: we can rotate, turn upside down the pieces[/color][/i]
Determine the number of real roots of the equation
$$16x^5 - 20x^3 + 5x + m = 0.$$
Find all integer pair $(m,n)$ such that $7^m=5^n+24$.
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and
gets the following answers:
$A_1$: On this island there is at least one scoundrel,
$A_2$: On this island there are at least two scoundrels,
$...$
$A_{n-1}$: On this island there are at least $n-1$ scoundrels,
$A_n$: On this island everybody is a scoundrel.
Can the journalist decide whether there are more scoundrels or more knights?
Let $d(n)$ be the number of positive integers that divide the integer $n$. For all positive integral divisors $k$ of $64800$, what is the sum of numbers $d(k)$?
$
\textbf{(A)}\ 1440
\qquad\textbf{(B)}\ 1650
\qquad\textbf{(C)}\ 1890
\qquad\textbf{(D)}\ 2010
\qquad\textbf{(E)}\ \text{None of above}
$
In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.
Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have
\[ \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 . \]
[i]Gabriel Dospinescu[/i]
If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$?
$\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$
Does there exist a two-variable polynomial $P(x, y)$ with real number coefficients such that $P(x, y)$ is positive exactly when $x$ and $y$ are both positive?
Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?
$\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) }
\frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}$
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD.
Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$.
Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$, $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ($c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$).
Prove: $\left| S - 10^k AB \right| \leq 9k.$