Found problems: 85335
Each cell of the checkered plane is colored in one of $n^2$ colors so that in any square of $n \times n$ cells all colors occur. It is known that in some line all the colors occur. Prove that there exists a column colored in exactly $n$ colors.
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]
A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that
$\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$
for any $x,y\not= 0$ and that
$\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$
for any $x\not\in\{ 0,1\}$. Determine all such functions $f$.
Let $x, y, z > -1$. Prove that \[ \frac{1+x^2}{1+y+z^2} + \frac{1+y^2}{1+z+x^2} + \frac{1+z^2}{1+x+y^2} \geq 2. \]
[i]Laurentiu Panaitopol[/i]
Give an isosceles triangle $ABC$ at $A$. Draw ray $Cx$ being perpendicular to $CA, BE$ perpendicular to $Cx$ ($E \in Cx$).Let $M$ be the midpoint of $BE$, and $D$ be the intersection point of $AM$ and $Cx$. Prove that $BD \perp BC$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following:
$$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$
[i]Proposed by Vadym Koval[/i]
Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that:
$ f(x)+f(\frac1x)=1$
$ 2f(f(x))=f(2x)$
Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.
Prove that the equation
$$\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}$$has finitely many solutions in positive integers.
Let $ABCD$ be a trapezium with $AD||BC$; and let $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $\angle DAY=\angle CAB$ then the internal bisectors of $\angle XAY$ and $\angle XBY$ meet on $XY$.
Proposed by Belur Jana Venkatachala
Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$
$ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$
The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$?
$ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$
Let $\triangle ABC$ be an acute triangle, $O$ be its circumcenter, and $D$ be a point different that $A$ and $C$ on the smaller $AC$ arc of its circumcircle. Let $P$ be a point on $[AB]$ satisfying $\widehat{ADP} = \widehat {OBC}$ and $Q$ be a point on $[BC]$ satisfying $\widehat{CDQ}=\widehat {OBA}$. Show that $\widehat {DPQ} = \widehat {DOC}$.
On the grid board shown, a token is placed on each white space.
[img]https://cdn.artofproblemsolving.com/attachments/3/2/0060b2436edb0ce25160d2f94f379defef237c.png[/img]
A move consists of choosing four squares on the board that form a "$T$" in any of the shapes shown below, and add a token to each of these four squares.
[img]https://cdn.artofproblemsolving.com/attachments/8/c/3890aed5289ec9ea2d147f8000a0422c233029.png[/img]
Is it possible, after carrying out several moves, to get the $25$ squares to have the same amount of chips?
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.
Given that $a_1, a_2, . . . , a_n$ are nonnegative real numbers with $a_1 + \cdots + a_n = 1$, prove that the expression
$$ \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} }$$
attains its minimum, and determine this minimum.
$ a,b,c$ are sides of triangle $ ABC$. For any choosen triple from $ (a \plus{} 1,b,c),(a,b \plus{} 1,c),(a,b,c \plus{} 1)$ there exist a triangle which sides are choosen triple. Find all possible values of area which triangle $ ABC$ can take.
Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.
In the expansion of $(x + b)^{2018}$, the coefficients of $x^2$ and $x^3$ are equal. Compute $b$.
Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true:
$$f(x+y+f(y))=f(x)+2y$$
[I]proposed by F. Petrov[/i]
Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]
Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.
Determine all triples $(a,b,c)$ of positive integers satisfying the conditions
$$\gcd(a,20) = b$$
$$\gcd(b,15) = c$$
$$\gcd(a,c) = 5$$
(Richard Henner)