Found problems: 85335
Let $ABCD$ be a convex cyclic quadrilateral satisfying $AB\cdot CD=AD\cdot BC$. Let the inscribed circle $\omega$ of triangle $ABC$ be tangent to sides $BC$, $CA$ and $AB$ at points $A', B'$ and $C'$, respectively. Let point $K$ be the intersection of line $ID$ and the nine-point circle of triangle $A'B'C'$ that is inside line segment $ID$. Let $S$ denote the centroid of triangle $A'B'C'$. Prove that lines $SK$ and $BB'$ intersect each other on circle $\omega$.
[i]Proposed by Áron Bán-Szabó, Budapest[/i]
Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.
For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$.
(1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$.
(2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis.
(3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$.
1992 Tokyo University entrance exam/Science, 2nd exam
We have an $n \times n$ board, and in every square there is an integer. The sum of all integers on the board is $0$. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists $n\geq 2$ such that we can turn all the integers into zeros in a finite number of actions.
The product of uncommon real roots of the two polynomials $ x^4 \plus{} 2x^3 \minus{} 8x^2 \minus{} 6x \plus{} 15$ and $ x^3 \plus{} 4x^2 \minus{} x \minus{} 10$ is ?
$\textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ \minus{} 6 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None}$
It is known that $a+\frac{b^2}{a}=b+\frac{a^2}{b}$. Is it true that $a=b$, where $a$ and $b$ are nonzero real numbers?
Proposed by R.Fedorov
Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells.
[i]Proposed by Merlijn Staps[/i]
Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that $\frac{M}{N}$ is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.
Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$
Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have
$$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$
Let $a0$, $a1$, ..., $a_n$ be integers, not all zero, and all at least $-1$. Given that $a_0+2a_1+2^2a_2+...+2^na_n =0$, prove that $a_0+a_1+...+a_n>0$.
The sports plane flew along a diamond-shaped route in windy weather. He flew through the first three sides of the rhombus in $a $, $b$ and $c$ hours, respectively. How long did it take him to cover the fourth side of the diamond? (The speed of an aircraft is a vector equal to the sum of two vectors: the aircraft’s own speed and the wind speed. Wind speed is a constant vector. The aircraft’s own speed is a vector of constant length).
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible.
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.
With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin$\dagger$ by which it was originally defeated. The number voting for the bill on the re-vote was $\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?
$\textbf{(A)}\ 75 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 20$
$\dagger$ In this context, margin of defeat (passage) is defined as the number of nays minus the number of ayes (nays-ayes).
By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
Consider the equation $$FORTY + TEN + TEN = SIXTY$$ , where each of the ten letters represents a distinct digit from $0$ to $9$. Find all possible values of $SIXTY$ .
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.
Let $ I,O $ denote the incenter, respectively, the circumcenter of a triangle $ ABC. $ The $ A\text{-excircle} $ touches the lines $ AB,AC,BC $ at $ K,L, $ respectively, $ M. $ The midpoint of $ KL $ lies on the circumcircle of $ ABC. $ Show that the points $ I,M,O $ are collinear.
[i]Павел Кожевников[/i]
In triangle $ABC$, $AD$ and $AH$ are the angle bisector and the altitude of vertex $A$, respectively. The perpendicular bisector of $AD$, intersects the semicircles with diameters $AB$ and $AC$ which are drawn outside triangle $ABC$ in $X$ and $Y$, respectively. Prove that the quadrilateral $XYDH$ is concyclic.
[i]Proposed by Mahan Malihi[/i]
Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational.
[i]Proposed by A. Golovanov[/i]
Four circles of radius $1$ with centers $A,B,C,D$ are in the plane in such a way that each circle is tangent to two others. A fifth circle passes through the center of two of the circles and is tangent to the other two.
Find the possible values of the area of the quadrilateral $ABCD$.
Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$, and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$. Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$, respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$, so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$. Prove that $DF,OM,EG$ are concurrent.
Suppose that there are $2001$ golf balls which are numbered from $1$ to $2001$ respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither $5$ nor $8$. How many such golf balls the box can contain at most? Justify your answer.