Found problems: 85335
A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation
$$ f(\sqrt{x^2 +y^2 })= f(x)f(y).$$
Prove that
$$f(x) =f(1)^{x^{2}}.$$
A two digit number $s$ is special if $s$ is the common two leading digits of the decimal expansion of $4^n$ and $5^n$, where $n$ is a certain positive integer. Given that there are two special number, find these two special numbers.
A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$. We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$. Determine the greatest value of $n$.
The first step in finding the product $ (3x \plus{} 2)(x \minus{} 5)$ by use of the distributive property in the form $ a(b \plus{} c) \equal{} ab \plus{} ac$ is:
$ \textbf{(A)}\ 3x^2 \minus{} 13x \minus{} 10 \qquad \textbf{(B)}\ 3x(x \minus{} 5) \plus{} 2(x \minus{} 5)\qquad \\\textbf{(C)}\ (3x \plus{} 2)x \plus{} (3x \plus{} 2)( \minus{} 5)\qquad \textbf{(D)}\ 3x^2 \minus{} 17x \minus{} 10\qquad \textbf{(E)}\ 3x^2 \plus{} 2x \minus{} 15x \minus{} 10$
Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.
There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.
Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.
Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)
What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$
Joyce made $12$ of her first $30$ shots in the first three games of this basketball game, so her seasonal shooting average was $40\% $. In her next game, she took $10$ shots and raised her seasonal shooting average to $50\% $. How many of these $10$ shots did she make?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$
Let $n\geq 1$ be an integer. Find all rings $(A,+,\cdot)$ such that all $x\in A\setminus\{0\}$ satisfy $x^{2^{n}+1}=1$.
We are swapping two different digits of a number in each step. If we start with the number $12345$, which of the following cannot be got after an even number of steps?
$
\textbf{(A)}\ 13425
\qquad\textbf{(B)}\ 21435
\qquad\textbf{(C)}\ 35142
\qquad\textbf{(D)}\ 43125
\qquad\textbf{(E)}\ 53124
$
Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that
\[ AB = \left(%
\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1 \\
\end{array}%
\right). \]
Find $BA$.
For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge2\sqrt{2xy}.$
A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\).
Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed.
[i]Proposed by Linus Tang[/i]
Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).
Let $n$ be a fixed positive integer. To any choice of real numbers satisfying \[0\le x_{i}\le 1,\quad i=1,2,\ldots, n,\] there corresponds the sum \[\sum_{1\le i<j\le n}|x_{i}-x_{j}|.\] Let $S(n)$ denote the largest possible value of this sum. Find $S(n)$.
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?
[asy]
size(4cm);
filldraw(scale(2)*unitcircle,gray,black);
filldraw(shift(-1,0)*unitcircle,white,black);
filldraw(shift(1,0)*unitcircle,white,black);
[/asy]
$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$
A function from the positive integers to the positive integers satisfies these properties
1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$.
2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$.
Prove that $f(2)=2, f(3)=3, f(1999)=1999$.
Call a (non-trivial) lattice class a pseudo-variety if it is closed under taking a homomorphic image, a direct product, and a convex subset. Prove that the smallest distributive pseudo-variety cannot be defined by a first-order set of formulas.
Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let$ BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?
Let $p > 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo 5.
Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called [i]compositional[/i] if
\[f(g(x)) \equiv x^{p^2} - x\]
in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs.
Two permutations $a_1,a_2,\dots,a_{2010}$ and $b_1,b_2,\dots,b_{2010}$ of the numbers $1,2,\dots,2010$ are said to [i]intersect[/i] if $a_k=b_k$ for some value of $k$ in the range $1\le k\le 2010$. Show that there exist $1006$ permutations of the numbers $1,2,\dots,2010$ such that any other such permutation is guaranteed to intersect at least one of these $1006$ permutations.
Find the greatest value of $p$ and the smallest value of $q$ such that for any triangle in the plane, the inequality
\[p<\frac{a+m}{b+n}<q\]
holds, where $a,b$ are it's two sides and $m,n$ their corresponding medians.
How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$.