Found problems: 260
Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.
Find all $f: R\rightarrow R$ such that
(i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite
(ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$
Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation.
[i]Proposed by Evan Chen[/i]
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]
You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.
Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$.
Evaluate $\int_0^2 g(t)dt$.
[i]2010 Kumamoto University entrance exam/Science[/i]
Let $k$ be a given positive integer. Define $x_{1}= 1$ and, for each $n > 1$, set $x_{n+1}$ to be the smallest positive integer not belonging to the set $\{x_{i}, x_{i}+ik | i = 1, . . . , n\}$. Prove that there is a real number $a$ such that $x_{n}= [an]$ for all $n \in\mathbb{ N}$.
Find the number of unordered pairs $\{ A,B \}$ of subsets of an n-element set $X$ that satisfies the following:
(a) $A \not= B$
(b) $A \cup B = X$
The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.
Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a \plus{} b$ is a perfect square, then $ p\left(a\right) \plus{} p\left(b\right)$ is also a perfect square.
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime?
b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?
Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying
\[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]
Circle $C$ (with center $O$) does not have common point with line $l$. Draw $OP$ perpendicular to $l$, $P \in l$. Let $Q$ be a point on $l$ ($Q$ is different from $P$), $QA$ and $QB$ are tangent to circle $C$, and intersect the circle at $A$ and $B$ respectively. $AB$ intersects $OP$ at $K$. $PM$, $PN$ are perpendicular to $QB$, $QA$, respectively, $M \in QB$, $N \in QA$. Prove that segment $KP$ is bisected by line $MN$.
Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively.
Prove that $ \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d}$ if $ ABCD$ has an incircle.
Extension: Prove that the converse is true, too.
[If this has already been posted, I humbly apologize. A quick search turned up nothing.]
Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$
[i]A. Hajnal[/i]
If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $\left| m - n \right|$ ?
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8,15,8,8,8,6,8,$ and $29$ as shown. All of its angles are right angles. Turn this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$.
[asy]
size(200);
defaultpen(linewidth(2));
pen qq=font("phvb");
pair rectangle[] = {origin,(0,-8),(15,-8),(15,-16),(23,-16),(23,-8),(29,-8),(29,0)};
string point[] = {"A","B","C","D","E","F","G","H"};
int dirlbl[] = {135,225,225,225,315,315,315,45};
string value[] = {"8","15","8","8","8","6","8","29"};
int direction[] = {0,90,0,90,180,90,180,270};
for(int i=0;i<=7;i=i+1)
{
draw(rectangle[i]--rectangle[(i+1) % 8]);
label(point[i],rectangle[i],dir(dirlbl[i]),qq);
label(value[i],(rectangle[i]+rectangle[(i+1) % 8])/2,dir(direction[i]),qq);
}
[/asy]
Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]
Let $ I$ be an ideal of the ring $ R$ and $ f$ a nonidentity permutation of the set $ \{ 1,2,\ldots, k \}$ for some $ k$. Suppose that for every $ 0 \not\equal{} a \in R, \;aI \not\equal{} 0$ and $ Ia \not\equal{}0$ hold; furthermore, for any elements $ x_1,x_2,\ldots ,x_k \in I$, \[ x_1x_2\ldots x_k\equal{}x_{1f}x_{2f}\ldots x_{kf}\] holds. Prove that $ R$ is commutative.
[i]R. Wiegandt[/i]
In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$.
We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle.
The following property is satisfied:
"for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element"
Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends.
([i]Serbia[/i])