Found problems: 85335
Consider the following inequality for real numbers $x,y,z$: $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ .
(a) Prove that the inequality is valid for $a = 2\sqrt2$
(b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$.
Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that:
$f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$
Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$
In a convex hexagon $ABCDEF$ the triangles $BDF, ACE$ are equilateral and congruent. Prove that the three lines connecting the midpoints of opposite sides are concurrent.
Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\] is a perfect cube.
Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that
$$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$
holds for all real numbers $x; y$
Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.
Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.
Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$.
[i]Proposed by John Murray, Ireland[/i]
As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]
Let $f$ be the function of the set of positive integers into itself, defi ned by $f(1) = 1$,
$f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the
number of positive odd integers m such that $f(m) = n$ is equal to the number of positive
integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$.
[color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]
Compute the number of ways to divide a $20 \times 24 $ rectangle into $4 \times 5$ rectangles. (Rotations and reflections are considered distinct.)
Solve the equation $x^2+1 = log_2(x+2)- 2x$.
In a forest, there are $n$ different trees (considered as points), no three of which lie on the same line. John takes photographs of the forest such that all trees are visible (and no two trees are behind each other). What is the largest number of orders of in which the trees that can appear on the photos?
[i]Proposed by Gábor Mészáros, Sunnyvale, Kalifornia[/i]
$x_1,x_2,\cdots,x_n$ are positive real numbers. Prove that
$$\frac{x_1^2}{x_2}+\frac{x_2^2}{x_3}+\cdots+\frac{x_n^2}{x_1}\geq x_1+x_2+\cdots x_n.$$
Let $ ABCD $ be a cyclic quadrilateral and let $ M $ be the set of incenters and excenters of the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $ (so 16 points in total). Prove that there exist two sets $ \mathcal{K} $ and $ \mathcal{L} $ of four parallel lines each, such that every line in $ \mathcal{K} \cup \mathcal{L} $ contains exactly four points of $ M $.
What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$?
$ \textbf{(A)}\ 49 \qquad
\textbf{(B)}\ 50 \qquad
\textbf{(C)}\ 51 \qquad
\textbf{(D)}\ 52 \qquad
\textbf{(E)}\ 53$
[b]p1.[/b] Given an equilateral triangle $ABC$ with area $16\sqrt3$, and an interior point $P$ with distances from vertices $|AP| = 4$ and $|BP| = 6$.
(a) Find the length of each side.
(b) Find the distance from point $P$ to the side $AB$.
(c) Find the distance $|PC|$.
[b]p2.[/b] Several players play the following game. They form a circle and each in turn tosses a fair coin. If the coin comes up heads, that player drops out of the game and the circle becomes smaller, if it comes up tails that player remains in the game until his or her next turn to toss. When only one player is left, he or she is the winner. For convenience let us name them $A$ (who tosses first), $B$ (second), $C$ (third, if there is a third), etc.
(a) If there are only two players, what is the probability that $A$ (the first) wins?
(b) If there are exactly $3$ players, what is the probability that $A$ (the first) wins?
(c) If there are exactly $3$ players, what is the probability that $B$ (the second) wins?
[b]p3.[/b] A circular castle of radius $r$ is surrounded by a circular moat of width $m$ ($m$ is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in the moat.
(a) If the radius $r$ is $140$ feet and there are only $3$ life guards available, what is the minimum possible width of moat they can watch?
(b) Find the minimum number of life guards needed as a function of $r$ and $m$.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/d7ff0e1227f9dcf7e49fe770f7dae928581943.png[/img]
[b]p4.[/b] (a)Find all linear (first degree or less) polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all linear polynomials $g(x)$.
(b) Prove your answer to part (a).
(c) Find all polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all polynomials $g(x)$.
(d) Prove your answer to part (c).
[b]p5.[/b] A non-empty set $B$ of integers has the following two properties:
i. each number $x$ in the set can be written as a sum $x = y+ z$ for some $y$ and $z$ in the set $B$. (Warning: $y$ and $z$ may or may not be distinct for a given $x$.)
ii. the number $0$ can not be written as a sum $0 = y + z$ for any $y$ and $z$ in the set $B$.
(a) Find such a set $B$ with exactly $6$ elements.
(b) Find such a set $B$ with exactly $6$ elements, and such that the sum of all the $6$ elements is $1988$.
(c) What is the smallest possible size of such a set $B$ ?
(d) Prove your answer to part (c).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find the index of the singular point $0$ of the vector field
\[(xy+yz+xz)\]
Let $a$, $b$, $c \in (0,+\infty)$ . Then the following inequality is true:$$\sqrt{(a+b)(b+c)}+\sqrt{(b+c)(c+a)}+\sqrt{(c+a)(a+b)}+a+b+c\leq \left(ab+bc+ca\right)\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)$$
Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\angle B = 90$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines $AH_c$ and $CH_a$ meet at point $K$.
Prove that $\angle MBK = 90$.
Find all natural numbers $n$ for which there is a permutation $\sigma$ of $\{1,2,\ldots, n\}$ that satisfies:
\[
\sum_{i=1}^n \sigma(i)(-2)^{i-1}=0
\]
Let $n$ and $k$ be positive integers, and let be the sets $X=\{1,2,3,...,n\}$ and $Y=\{1,2,3,...,k\}$.
Let $P$ be the set of all the subsets of the set $X$. Find the number of functions $ f: P \to Y$ that satisfy $f(A \cap B)=\min(f(A),f(B))$ for all $A,B \in P$.
There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted:
(i) Remove an equal number of balls from both boxes.
(ii) Increase the number of balls in one of the boxes by a factor $k$.
Is it possible to remove all of the balls from both boxes with just these two actions,
1. if $k = 2$?
2. if $k = 3$?
Let n be the natural number ($n\ge 2$). All natural numbers from $1$ up to $n$ ,inclusive, are written on the board in some order: $a_1$, $a_2$ , $...$ , $a_n$. Determine all natural numbers $n$ ($n\ge 2$), for which the product
$$P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n)$$
is an even number, whatever the arrangement of the numbers written on the board.