Found problems: 85335
Given a rational number $a \neq 0$, find all functions $f:\mathbb{Q} \rightarrow \mathbb{Q}$ such that $$f(f(x)+ay)=af(y)+x$$ for all rational $x, y$.
A section of a rectangular parallelepiped by a plane is a regular hexagon. Prove that this parallelepiped is a cube.
Consider the following layouts of nine triangles with the letters $A, B, C, D, E, F, G, H, I$ in its interior.
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(200);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = 1.740000000000003, xmax = 8.400000000000013, ymin = 3.500000000000005, ymax = 9.360000000000012; /* image dimensions */
draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)--cycle);
/* draw figures */
draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005));
draw((2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286));
draw((7.461947712046029,4.569577506690286)--(5.020000000000005,8.820000000000011));
draw((3.382989341689345,5.990838871467448)--(4.193333333333338,4.580000000000005));
draw((4.202511849578174,7.405966442513598)--(5.828619600041468,4.573707435672692));
draw((5.841878190157451,7.408513542990484)--(4.193333333333338,4.580000000000005));
draw((6.656214943659867,5.990342259816768)--(5.828619600041468,4.573707435672692));
draw((4.202511849578174,7.405966442513598)--(5.841878190157451,7.408513542990484));
draw((3.382989341689345,5.990838871467448)--(6.656214943659867,5.990342259816768));
label("\textbf{A}",(4.840000000000007,8.020000000000010),SE*labelscalefactor,fontsize(22));
label("\textbf{B}",(3.980000000000006,6.640000000000009),SE*labelscalefactor,fontsize(22));
label("\textbf{C}",(4.820000000000007,7.000000000000010),SE*labelscalefactor,fontsize(22));
label("\textbf{D}",(5.660000000000008,6.580000000000008),SE*labelscalefactor,fontsize(22));
label("\textbf{E}",(3.160000000000005,5.180000000000006),SE*labelscalefactor,fontsize(22));
label("\textbf{F}",(4.020000000000006,5.600000000000008),SE*labelscalefactor,fontsize(22));
label("\textbf{G}",(4.800000000000007,5.200000000000007),SE*labelscalefactor,fontsize(22));
label("\textbf{H}",(5.680000000000009,5.620000000000007),SE*labelscalefactor,fontsize(22));
label("\textbf{I}",(6.460000000000010,5.140000000000006),SE*labelscalefactor,fontsize(22));
/* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
A sequence of letters, each letter chosen from$ A, B, C, D, E, F, G, H, I$ is said to be [i]triangle-friendly[/i] if the first and last letter of the sequence is $C$, and for every letter except the first letter, the triangle containing this letter shares an edge with the triangle containing the previous letter in the sequence. For example, the letter after $C$ must be either $A, B$, or $D$. For example, $CBF BC$ is triangle-friendly, but $CBF GH$ and $CBBHC$ are not.
[list]
[*] (a) Determine the number of triangle-friendly sequences with $2012$ letters.
[*] (b) Determine the number of triangle-friendly sequences with exactly $2013$ letters.[/list]
Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?
Find all real roots of $f$ if $f(x^{1/9})=x^2-3x-4$.
Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.
Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?
[i]Also compare shortlist 2009, combinatorics problem C1.[/i]
With $21$ tiles, some white and some black, a $3 \times 7$ rectangle is formed. Show that there are always four tokens of the same color located at the vertices of a rectangle.
For all real numbers $x$, let $P(x)=16x^3 - 21x$. What is the sum of all possible values of $\tan^2\theta$, given that $\theta$ is an angle satisfying \[P(\sin\theta) = P(\cos\theta)?\]
Solve the system
$\begin{cases}
x^2y^2+|xy|=\frac{4}{9}\\
xy+1=x+y^2\\
\end{cases}$
How many positive integer multiples of 1001 can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?
An arrangement of 12 real numbers in a row is called [i]good[/i] if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the third?
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
The altitudes $AD$ and $BE$ of acute triangle $ABC$ intersect at $H$. Let $F$ be the intersection of $AB$ and a line that is parallel to the side $BC$ and goes through the circumcentre of $ABC$. Let $M$ be the midpoint of $AH$. Prove that $\angle CMF=90^\circ$
Let $p$ be an odd prime number. Prove that
$$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$
Find a sequence of $ 2012 $ distinct integers bigger than $ 0 $ such that their sum is a perfect square and their product is a perfect cube.
A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.
(i) The value of $a_0$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\frac{a_i}{a_i + 2} $
(iii) There exists a positive integer $k$ such that $a_k = 2014$.
Find the smallest positive integer $n$ such that there exists a good sequence $a_0, a_1, . . .$ of real numbers with the property that $a_n = 2014$.
[i]Proposed by Wang Wei Hua, Hong Kong[/i]
It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?
Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
$$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$
is the sphere
$$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$.
Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.
[b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.
In an acute-angled triangle $ABC$, the circle $S$ with the altitude $BK$ as the diameter intersects $AB$ at $E$ and $BC$ at $F$. Prove that the tangents to $S$ at $E$ and $F$ meet on the median from $B$.
Determine the smallest positive integer $n \ge 3$ for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$).
Let $AB$ be a diameter of a circle $\Gamma$ with center $O$. Let $CD$ be a chord where $CD$ is perpendicular to $AB$, and $E$ is the midpoint of $CO$. The line $AE$ cuts $\Gamma$ in the point $F$, the segment $BC$ cuts $AF$ and $DF$ in $M$ and $N$, respectively. The circumcircle of $DMN$ intersects $\Gamma$ in the point $K$. Prove that $KM=MB$.
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.
A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:
$\textbf{(A)}\ \text{Least when the point is the center of gravity of the triangle}\qquad\\
\textbf{(B)}\ \text{Greater than the altitude of the triangle} \qquad\\
\textbf{(C)}\ \text{Equal to the altitude of the triangle}\qquad\\
\textbf{(D)}\ \text{One-half the sum of the sides of the triangle} \qquad\\
\textbf{(E)}\ \text{Greatest when the point is the center of gravity}$