Found problems: 85335
For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is
$ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$
Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.
[i]L. Kuptsov[/i]
Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that
\[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]
The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.
Each number of the set $\{1,2, 3,4,5,6, 7,8\}$ is colored red or blue, following the following rules:
(a) Number $4$ is colored red, and there is at least one blue number,
(b) if two numbers $x,y$ have different colors and $x + y \le 8$, so the number $x + y$ is colored blue,
(c) if two numbers $x,y$ have different colors and $x \cdot y \le 8$, then the number $x \cdot y$ is colored red.
Find all the possible ways to color this set.
Let $A=\{1,3,3^2,\ldots, 3^{2014}\}$. We obtain a partition of $A$ if $A$ is written as a disjoint union of nonempty subsets.
[list=a]
[*]Prove that there is no partition of $A$ such that the product of elements in each subset is a square.
[*]Prove that there exists a partition of $A$ such that the sum of elements in each subset is a square.[/list]
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
Find the total area of the shaded area in the figure if all circles have an equal radius $R$ and the centers of the outer circles divide into six equal parts of the middle circle.
[img]http://3.bp.blogspot.com/-Ax0QJ38poYU/XovXkdaM-3I/AAAAAAAALvM/DAZGVV7TQjEnSf2y1mbnse8lL6YIg-BQgCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.5.png[/img]
Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$
$$
\begin{cases}
a_{n+1}=7a_n+6b_n-3, \\
b_{n+1}=8a_n+7b_n-4.
\end{cases}
$$
Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.
A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.
How many ways can you color a necklace of $7$ beads with $4$ colors so that no two adjacent beads have the same color?
a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions
(1) $ABCD$ is not cyclic;
(2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths;
(3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal?
b) Does there exist such a non-convex quadrangle?
Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations
$$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\
g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$
for all $ x,y\in\mathbb{Q} . $
Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.
The incircle of triangle $ABC$ touches the sides $BC,CA,AB$ at the points $D,E,F$, respectively. Let $K$ be a point on the side $BC$ and let $M$ be the point on the line segment $AK$ such that $AM=AE=AF$. Denote by $L,N$ the incenters of triangles $ABK,ACK$, respectively.
Prove that $K$ is the foot of the altitude from $A$ if and only if $DLMN$ is a square.
[hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2[/url]
[/hide]
[i]Bogdan Enescu[/i]
Let $ABCD$ be a rectangle. The point $E$ lies on side $ \overline{AB}$ and the point $F$ is lies side $ \overline{AD}$, such that $\angle FEC=\angle CEB$ and $\angle DFC=\angle CFE$. Determine the measure of the angle $\angle FCE$ and the ratio $AD/AB$.
Let the circle $\omega$ be circumscribed around the triangle $\vartriangle ABC$ with right angle $\angle A$. Tangent to the circle $\omega$ at point $A$ intersects the line $BC$ at point $D$. Point $E$ is symmetric to $A$ with respect to the line $BC$. Let $K$ be the foot of the perpendicular drawn from point $A$ on $BE$, $L$ the midpoint of $AK$. The line $BL$ intersects the circle $\omega$ for the second time at the point $N$. Prove that the line $BD$ is tangent to the circle circumscribed around the triangle $\vartriangle ADM$.
Rama was asked by her teacher to subtract $3$ from a certain number and then divide the result by $9$. Instead, she subtracted $9$ and then divided the result by $3$. She got $43$ as the answer. What should have been her answer if she had solved the problem correctly?
How many possible values are there for the sum
a + b + c + d if a; b; c; d are positive integers and abcd = 2007:
Find all positive integers $n$ for which the equation $$x^3 + y^3 = n! + 4$$ has solutions in integers.
The ratio of the interior angles of two regular polygons with sides of unit length is $ 3: 2$. How many such pairs are there?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ \text{infinitely many}$
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that
$$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$
When is the equality valid?