Found problems: 85335
Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$.
Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.
Prove that all continuous solutions of the functional equation
$\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)$
are the constant functions.
Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.
$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?
Eight points are chosen on the circumference of a circle, labelled $P_1$, $P_2$, ..., $P_8$ in clockwise order. A route is a sequence of at least two points $P_{a_1}$, $P_{a_2}$, $...$, $P_{a_n}$ such that if an ant were to visit these points in their given order, starting at $P_{a_1}$ and ending at $P_{a_n}$, by following $n-1$ straight line segments (each connecting each $P_{a_i}$ and $P_{a_{i+1}}$), it would never visit a point twice or cross its own path. Find the number of routes.
Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\
bc + d + a = 5 \\
cd + a + b = 2 \\
da + b + c = 6 \end{cases}$
Given a prime number $p$, find all integer solutions of $p(x+y) = xy$.
Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.
Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized.
\[ \int_0^{\pi} |\sin x-ax|\ dx \]
Let $ t \ge 3$ be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)\minus{}t^k|<1$, for $ k\equal{}0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.
In the convex quadrilateral $ABCD, BC$ is parallel to $AD$. Two circular arcs $\omega_1$ and $\omega_3$ pass through $A$ and $B$ and are on the same side of $AB$. Two circular arcs $\omega_2$ and $\omega_4$ pass through $C$ and $D$ and are on the same side of $CD$. The measures of $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are $\alpha, \beta,\beta$ and $\alpha$ respectively. If $\omega_1$ and $\omega_2$ are tangent to each other externally, prove that so are $\omega_3$ and $\omega_4$.
Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$.
[i]Proposed by Alex Li[/i]
Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions:
1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square.
2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square.
Find the maximum value of the number of the “good square”.
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that
(i) $a+b+c+d=n$
(ii) $a \ge b \ge d$
(iii) $a \ge c \ge d$
Prove that for all $n$, $p(n) = q(n)$.
In a country 6 cities are connected two by two with round-trip air routes operated by exactly one of the two air companies in that country.
Prove that there exist 4 cities $A$, $B$, $C$ and $D$ such that each of the routes $A\leftrightarrow B$, $B\leftrightarrow C$, $C\leftrightarrow D$ and $D\leftrightarrow A$ are operated by the same company.
[i]Dan Schwartz[/i]
A positive integer is [i]happy[/i] if:
1. All its digits are different and not $0$,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a [i]happy[/i] number. How many [i]happy[/i] numbers are there?
Let n be a positive interger. Let n real numbers be wrote on a paper. We call a "transformation" :choosing 2 numbers $a,b$ and replace both of them with $a*b$. Find all n for which after a finite number of transformations and any n real numbers, we can have the same number written n times on the paper.
Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .
Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that:
\[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}}
\]
A man on his way to dinner short after $ 6: 00$ p.m. observes that the hands of his watch form an angle of $ 110^{\circ}.$ Returning before $ 7: 00$ p.m. he notices that again the hands of his watch form an angle of $ 110^{\circ}.$ The number of minutes that he has been away is:
$ \textbf{(A)}\ 36 \frac23 \qquad
\textbf{(B)}\ 40 \qquad
\textbf{(C)}\ 42 \qquad
\textbf{(D)}\ 42.4 \qquad
\textbf{(E)}\ 45$
Let $ABC$ be a triangle with $\angle BAC=40^\circ $, $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$. If $AD\parallel OG$, find $\angle ACB$.
Consider $n \ge 2$ pairwise disjoint disks $D_1,D_2,\ldots,D_n$ on the Euclidean plane. For each $k=1,2,\ldots,n$, denote by $f_k$ the inversion with respect to the boundary circle of $D_k$. (Here, $f_k$ is defined at every point of the plane, except for the center of $D_k$.) How many fixed points can the transformation $f_n\circ f_{n-1}\circ\ldots\circ f_1$ have, if it is defined on the largest possible subset of the plane?
Find all triples $(x,y,z)$ of positive integers such that
$$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$
\[\int_0^1 \frac{(x+1)\log(x)}{x^3-1}\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]