Practice (125)
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Show that the following inequality holds for any positive integer $n$: $$(2n+1)^n \ge (2n)^n + (2n-1)^n$$
Let $a$ and $b$ be two positive real numbers. Show that if $\frac{1}{a}+\frac{1}{b}=1$. Prove that the following inequality holds for any positive integer $n$: $$(a+b)^n-a^n-b^n\ge 2^{2n}-2^{n+1}$$
Let $a$, $b$ be two positive real numbers, and $n$ be a positive integer greater than $2$. Show that $$\frac{a^n+a^{n-1}b+\cdots+ab^{-1}+b^n}{n+1}\ge \Big(\frac{a+b}{2}\Big)^n$$
Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.
Let positive integers $a_1, a_2, \cdots, a_{31}, b_1, b_2, \cdots, b_{31}$ satisfy the following conditions:
- $a_1 < a_2 < a_3 < \cdots < a_{31}$
- $b_1 < b_2 < b_3 < \cdots < b_{31}$
- $a_1 + a_2+a_3+\cdots + a_{31} = b_1 + b_2 + b_3 + \cdots + b_{31}=2015$
Find the maximum value of $S=\mid a_1 - b_1 \mid + \mid a_2 - b_2 \mid + \cdots + \mid a_{31}-b_{31}\mid$.
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
Find the largest integer not exceeding $1 + \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}} + \cdots + + \frac{1}{\sqrt{10000}}$
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$
All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
Let $b$ and $c$ be two given real numbers. What is the maximum number of distinct integers $x$ are there such that $$\mid 101x^2 + bx + c\mid \le 50$$?
Let $G$ be the centroid of $\triangle{ABC}$. Points $M$ and $N$ are on side $AB$ and $AC$, respectively such that $\overline{AM} = m\cdot\overline{AB}$ and $\overline{AN} = n\cdot\overline{AC}$ where $m$ and $n$ are two positive real numbers. Find the minimal value of $mn$.
How many distinct isosceles triangles having sides of integral lengths and perimeter 113 are possible?
There are four points on a plane as shown. Points $A$ and $B$ are fixed points satisfying $AB=\sqrt{3}$. Points $P$ and $Q$ can move, as long as $AP=PQ=QB=1$. Let $S$ and $T$ be the area of $\triangle{APB}$ and $\triangle{PQB}$, respectively. Find the maximum value of $S^2+T^2$.
Squares $ABCD$ and $EFGH$ have a common center at $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.
(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$
Let real numbers $x_1$ and $x_2$ satisfy $ \frac{\pi}{2} > x_1 > x_2 > 0$, show $$\frac{\tan x_1}{x_1} > \frac{\tan x_2}{x_2}$$
As shown.
Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A, B,$ and $C$ are integers satisfying $A\log16+B\log18+C\log24=0$.
In $\triangle{LEO}$, point $J$ lies on $\overline{LO}$ such that $\overline{JE}\perp\overline{EO}$, and point $S$ lies on $\overline{LE}$ such that $\overline{JS}\perp\overline{LE}$. Given that $JS=9, EO=20,$ and $JO+SE=37$, compute the perimeter of $\triangle{LEO}$.
Compute the least possible area of a non-degenerate right triangle with sides of lengths $\sin{x}$, $\cos{x}$ and $\tan{x}$ where $x$ is a real number.
During the annual frog jumping contest at the county fair, the height of the frog's jump, in feet, is given by $$f(x)= -
\frac{1}{3}x^2+\frac{4}{3}x\:$$ What was the maximum height reached by the frog?
Let $x, y,$ and $z$ be some real numbers such that: $x+2y-z=6$ and $x-y+2z=3$. Find the minimal value of $x^2 + y^2 + z^2$.
Let $x$ be a negative real number. Find the maximum value of $y=x+\frac{4}{x} +2007$.
Let real numbers $a$ and $b$ satisfy $a^2 + ab + b^2 = 1$. Find the range of $a^2 - ab + b^2$.