Found problems: 85335
Three married couples arrange a party. They arrive at the party one at a time, the couples not necessarily arriving together. They all, upon arriving, shake the hand of everyone already there, except their own spouse. When everyone has arrived, someone asks all the others how many hands they shook upon arriving, and gets five different answers. How many hands did he himself shake upon arriving?
A. 0
B. 1
C. 2
D. 3
E. 4
Find all integers $b$ and $c$ for which the equation $x^2-bx+c=0$ has two real roots $x_{1}$ and $x_{2}$ satisfying $x_{1}^2+x_{2}^2=5$.
Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.
Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds
$$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$
The triangle $ABC$ is inscribed in a circle $\gamma$ of center $O$, with $AB < AC$ . A point $D$ on the angle bisector of $\angle BAC$ and a point $E$ on segment $BC$ satisfy $OE$ is parallel to $AD$ and $DE \perp BC$. Point $K$ lies on the extension line of $EB$ such that $EA = EK$. A circle pass through points $A,K,D$ meets the extension line of $BC$ at point $P$, and meets the circle of center $O$ at point $Q\ne A$. Prove that the line $PQ$ is tangent to the circle $\gamma$.
[5] Find all integers $n$ for which $\frac{n^3+8}{n^2-4}$ is an integer.
Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.
Let be three nonzero rational numbers $ a,b,c $ under the relation $ (a+b)(b+c)(c+a)=a^2b^2c^2. $
Show that the expression $ \sqrt[3]{3+1/a^3+1/b^3+1/c^3} $ is rational.
[i]Ion Bursuc[/i]
There is a point $D$ on side $AC$ of acute triangle $\triangle ABC$. Let $AM$ be the median drawn from $A$ (so $M$ is on $BC$) and $CH$ be the altitude drawn from $C$ (so $H$ is on $AB$). Let $I$ be the intersection of $AM$ and $CH$, and let $K$ be the intersection of $AM$ and line segment $BD$. We know that $AK=8$, $BK=8$, and $MK=6$. Find the length of $AI$.
A positive integer $\ell \ge 2$ is called [i]sweet[/i] if there exists a positive integer $n \ge 10$ such that when the leftmost nonzero decimal digit of $n$ is deleted, the resulting number $m$ satisfies $n = m\ell.$ Let $S$ denote the set of all sweet numbers $\ell.$ If the sum $\sum_{\ell \in S} \tfrac{1}{\ell-1}$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$
Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$ and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$ , $n\geq 1$.
Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n}x_k$ is a perfect square.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.
In $\triangle ABC$, $AB>AC$, $l$ is tangent line of the circumscribed circle of $\triangle ABC$ that passes $A$. The circle with center $A$ and radius $AC$, intersects segment $AB$ at $D$, and line $l$ at $E, F$ ($F,B$ are on the same side). Prove that lines $DE, DF$ pass the incenter and an excenter of $\triangle ABC$ respectively.
a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines).
b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.
The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this:
[img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img]
The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .
Point $X$ lies in a right-angled isosceles $\triangle ABC$ ($\angle ABC = 90^\circ$). Prove that
$AX+BX+\sqrt{2}CX \geq \sqrt{5}AB$
and find for which points $X$ the equality is met.
Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that
\[ f(x+f(y)) = yf(xy+1)\]
holds for all $ x, y \in \mathbb{R} ^{+} $.
Let $ABCD$ be a convex quadrilateral with $AC \perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the distance from $P$ to $AB$ is $99$, the distance from $P$ to $BC$ is $63$, and the distance from $P$ to $CD$ is $77$. What is the distance from $P$ to $AD$?
Let $ABC$ be an obtuse triangle, and let $H$ denote its orthocenter. Let $\omega_A$ denote the circle with center $A$ and radius $AH$. Let $\omega_B$ and $\omega_C$ be defined in a similar way. For all points $X$ in the plane of triangle $ABC$ let circle $\Omega(X)$ be defined in the following way (if possible): take the polars of point $X$ with respect to circles $\omega_A$, $\omega_B$ and $\omega_C$, and let $\Omega(X)$ be the circumcircle of the triangle defined by these three lines.
With a possible exception of finitely many points find the locus of points $X$ for which point $X$ lies on circle $\Omega(X)$.
[i]Proposed by Vilmos Molnár-Szabó, Budapest[/i]
On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $
Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.
[b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$.
[b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that
\[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\]
if and only if $ABC$ is acute-angled.
Let $ a$ be a real number. Prove that if the equation $ x^2\minus{}ax\plus{}a\equal{}0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).$
Let $I$ be the incenter of triangle $ABC$. Let $BI$ and $AC$ intersect at $E$, and $CI$ and $AB$ intersect at $F$. Suppose that $R$ is another intersection of $\odot (ABC)$ and $\odot (AEF)$. Let $M$ be the midpoint of $BC$, and $P, Q$ are the intersections of $AI, MI$ and $EF$, respectively. Show that $A, P, Q, R$ are concyclic.
(ltf0501).
Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$?
$\text{(A)}$ It has no real roots.
$\text{(B)}$ It has two equal real roots.
$\text{(C)}$ It has two different real roots, and their product is positive.
$\text{(D)}$ It has two different real roots, and their product is negative.