Found problems: 85335
Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x \rfloor$.
Let $ABC$ be an acute triangle.
Find the locus of the centers of the rectangles which have their vertices on the sides of $ABC$.
Suppose $p$ is a prime with $p \equiv 3 \; \pmod{4}$. Show that for any set of $p-1$ consecutive integers, the set cannot be divided two subsets so that the product of the members of the one set is equal to the product of the members of the other set.
On the puzzle page of a newspaper this problem is proposed:
“Two children, Antonio and José, have $160$ comics. Antonio counts his by $7$ by $7$ and there are $4$ left over. José counts his $ 8$ by $8$ and he also has $4$ left over. How many comics does he have each?" In the next issue of the newspaper this solution is given: “Antonio has $60$ comics and José has $100$.”
Analyze this solution and indicate what a mathematician would do with this problem.
Circles $S_1{}$ and $S_2{}$ intersect in $M{}$ and $N{}$. Line $l$ intersects the circles in points $A,B\in S_1$ and $C,D\in S_2$. Prove that $\angle AMC=\angle BND$ and $\angle ANC=\angle BMD$ if the order of points on line $l$ is:
[b]a)[/b] $A,C,B,D;\quad$ [b]b)[/b] $A,C,D,B.$
Find the set of solutions to the equation
$\log_{\lfloor x\rfloor}(x^2-1)=2$
On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$, $IJ=3$, $JK=4$, $KH=5$. Find the value of $13(BD)^2$.
There are 111 StarCraft programmers. The StarCraft team SKT starts with a given set of eleven programmers on it, and at the end of each season, it drops a progamer and adds a programmer (possibly the same one). At the start of the second season, SKT has to fi eld a team of five programmers to play the opening match. How many diff erent lineups of ve players could be fi elded if the order of players on the lineup matters?
For the family of parabolas
$$y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a$$
(i) find the locus of vertices,
(ii) find the envelope,
(iii) sketch the envelope and two typical curves of the family.
There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$. the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$. There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines.
Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.
Find all natural numbers $x,y$ such that $$x^5=y^5+10y^2+20y+1.$$
Define a sequence of polynomials $F_n(x)$ by $F_0(x)=0, F_1(x)=x-1$, and for $n\geq 1$, $$F_{n+1}(x)=2xF_n(x)-F_{n-1}(x)+2F_1(x).$$ For each $n$, $F_n(x)$ can be written in the form $$F_n(x)=c_nP_1(x)P_2(x)\cdots P_{g(n)}(x)$$ where $c_n$ is a constant and $P_1(x),P_2(x)\cdots, P_{g(n)}(x)$ are non-constant polynomials with integer coefficients and $g(n)$ is as large as possible. For all $2< n< 101$, let $t$ be the minimum possible value of $g(n)$ in the above expression; for how many $k$ in the specified range is $g(k)=t$?
Let $ABCD$ be a trapezoid such that $|AB|=9$, $|CD|=5$ and $BC\parallel AD$. Let the internal angle bisector of angle $D$ meet the internal angle bisectors of angles $A$ and $C$ at $M$ and $N$, respectively. Let the internal angle bisector of angle $B$ meet the internal angle bisectors of angles $A$ and $C$ at $L$ and $K$, respectively. If $K$ is on $[AD]$ and $\dfrac{|LM|}{|KN|} = \dfrac 37$, what is $\dfrac{|MN|}{|KL|}$?
$
\textbf{(A)}\ \dfrac{62}{63}
\qquad\textbf{(B)}\ \dfrac{27}{35}
\qquad\textbf{(C)}\ \dfrac{2}{3}
\qquad\textbf{(D)}\ \dfrac{5}{21}
\qquad\textbf{(E)}\ \dfrac{24}{63}
$
$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$.
[i]Proposed by Alex Li[/i]
Euler is selling Mathematician cards to Gauss. Three Fermat cards plus $5$ Newton cards costs $95$ Euros, while $5$ Fermat cards plus $2$ Newton cards also costs $95$ Euros. How many Euroes does one Fermat card cost?
$$
\mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35
$$
One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was
$ \text{(A)}\ \text{96 cents}\qquad\text{(B)}\ \text{1.07 dollars}\qquad\text{(C)}\ \text{1.18 dollars}\qquad\text{(D)}\ \text{1.20 dollars}\qquad\text{(E)}\ \text{1.40 dollars} $
Suppose $f : [0,1] \longrightarrow \mathbb{R}$ is differentiable with $f(0) = 0$. If $|f'(x) | \leq f(x)$ for all $x \in [0,1]$, then show that $f(x) = 0$ for all $x$.
For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of
\[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]
Call two regular polygons supplementary if the sum of an internal angle from each polygon adds up to $180^o$. For instance, two squares are supplementary because the sum of the internal angles is $90^o + 90^o = 180^o$. Find the other pair of supplementary polygons. Write your answer in the form $(m, n)$ where m and n are the number of sides of the polygons and $m < n$.
Find all pairs of natural numbers $(p,n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$
Find all integral solutions of the equation $$|p^r-q^s|=1,$$ where $p$ and $q$ are prime numbers and $r$ and $s$ are positive integers larger than unity. Prove that there are no other solutions.
Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$
A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the $27$ row sums can be odd?
Let $A=[a_{ij}]_{n \times n}$ be a $n \times n$ matrix whose elements are all numbers which belong to set $\{1,2,\cdots ,n\}$. Prove that by swapping the columns of $A$ with each other we can produce matrix $B=[b_{ij}]_{n \times n}$ such that $K(B) \le n$ where $K(B)$ is the number of elements of set $\{(i,j) ; b_{ij} =j\}$.
In the Intergalactic Lottery, $7$ numbers are drawn out of $55$. R2-D2 and C-3PO decide that they want to win this lottery, so they fill out lottery tickets separately such that for each possible draw one of them does have a winning ticket for that draw. Prove that one of them has $7$ tickets with all different numbers.