Found problems: 85335
Let the incircle of triangle $ABC$ meet the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, and let the $A$-excircle meet the lines $BC$, $CA$, $AB$ at $P$, $Q$, $R$. Let the line passing through $A$ and perpendicular to $BC$ meet the lines $EF$, $QR$ at $K$, $L$. Let the intersection of $LD$ and $EF$ be $S$, and the intersection of $KP$ and $QR$ be $T$. Prove that $A$, $S$, $T$ are collinear.
How many ordered positive integer triples $(x,y,z)$ are there such that
\[
\begin{array}{rcl}
3x^2-2y^2-4z^2+54 = 0 \\
5x^2-3y^2-7z^2+74 = 0
\end{array}
\]
$ \textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely Many}
\qquad\textbf{(E)}\ \text{None of the options}
$
[b]8.[/b] Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. [b](G. 10)[/b]
Is it possible to cut a regular triangular prism into two equal pyramids?
Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B',D',E'$ and $C'$ lie on the same circle.
The product of two positive numbers is $9$. The reciprocal of one of these numbers is $4$ times the reciprocal of the other number. What is the sum of the two numbers?
$ \textbf{(A)}\ \dfrac{10}{3}
\qquad\textbf{(B)}\ \dfrac{20}{3}
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ \dfrac{15}{2}
\qquad\textbf{(E)}\ 8
$
Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.
Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that
[i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$
[i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$
Prove that if the numbers $a, b, c$ are related by the relation $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}= \frac{1}{a+b+c}$ then the sum of some two of them is equal to zero.
Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
Prove that there is no polynomial $P(x)$ with integer coefficients such that
$$P(\sqrt[3]{5} + \sqrt[3]{25}) = 2\sqrt[3]{5}+3\sqrt[3]{25}$$
Show that there are no triples $(a, b, c)$ of positive integers such that
a) $a + c, b + c, a + b$ do not have common multiples in pairs.
b)$\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.
Let $\mathcal{G}$ be the set of all points $(x,y)$ in the Cartesian plane such that $0\le y\le 8$ and $$(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.$$ There exists a unique line $\ell$ of [b]negative slope[/b] tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a [b]unique[/b] point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$
Let $f$ be a real function such that for all $x\ne 0$, $x\ne 1$,
$$f (x) + f \left(- \frac{1}{x - 1} \right) =\frac{9}{4x^2} + f\left(1 - \frac{1}{x} \right) .$$
Compute $f \left( \frac{1}{2}\right).$
.
Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.
Bisectors of a right triangle $\triangle ABC$ with right angle $B$ meet at point $I.$ The perpendicular to $IC$ drawn from $B$ meets the line $IA$ at $D;$ the perpendicular to $IA$ drawn from $B$ meets the line $IC$ at $E.$ Prove that the circumcenter of the triangle $\triangle IDE$ lies on the line $AC.$
[i](A. Kuznetsov )[/i]
Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$, in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$, there exists a box containing $n$ ballots. For each $n$, find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.
Calculate the following indefinite integrals.
[1] $\int \sin (\ln x)dx$
[2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$
[3] $\int \frac{x^3}{x^2+1}dx$
[4] $\int \frac{x^2}{\sqrt{2x-1}}dx$
[5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$
The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game :
each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?
For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$
Find the value of the expression $x^2 + y^2+ z^2$.
Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.
$P$ and $Q$ are points on angle bisectors of two adjacent angles. Let $PA$, $PB$, $QC$ and $QD$ be altitudes on the sides of these adjacent angles. Prove that lines $AB$, $CD$ and $PQ$ are concurrent.