Found problems: 85335
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \dots,20$ on its sides). He conceals the results but tells you that at least half the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?
Find the unique integer $a > 1$ that satisfies
\[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]
Find all solutions of the system
$\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $
$\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $
$\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$
$x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$
where $x, y, z \ge 0$ and $w \in R$
[hide=PS] I attached the system, in case I have any typos[/hide]
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer.
Prove that $M(a,b)$ is a square.
(b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.
Find the number of ordered triplets $(a,b,c)$ of positive integers such that $a<b<c$ and $abc=2008$.
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$.
[i]Alex Zhu.[/i]
Let $E$ be the ellipse lying in the $x, y$ plane centered at $(0, 0)$ with semi-major axis of length $2$ along the $x$-axis and semi-minor axis of length $1$ along the $y$-axis. Let $C$ be a cone created by revolving two perpendicular lines about an angle bisector of the perpendicular angle. There are some points $(x, y, z)$ where the vertex of $C$ could be so that $E$ is the intersection of $C$ with the $x, y$ plane. These points define a convex polygon in the $x, z$ plane. The area of this polygon can be expressed as $\sqrt{n}$ for some positive integer $n.$ Find $n.$
(Some definitions: the semi-major axis is the longest distance from the center of the ellipse to the boundary, and the semi-minor axis is the shortest distance from the center of the ellipse to the boundary.)
The value of $ x \plus{} x(x^x)$ when $ x \equal{} 2$ is:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 36 \qquad\textbf{(E)}\ 64$
What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping?
(A) $Mg$
(B) $\mu_k Mg$
(C) $\mu_k Mg/\mu_s$
(D) $(\mu_s + \mu_k)Mg$
(E) $Mg/\mu_s$
Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$.
Nguyễn Duy Thái Sơn
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?
Evaluate the integral $\int_0^{\pi/2} \sqrt{\tan \theta} d\theta$.
Determine all real polynomials $P$ of degree at most $22$ for which
$$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$
for all $k \in \{1, 2, 3, . . . , 21, 22\}$.
Prove that every polynomial of fourth degree can be represented in the form $P(Q(x))+R(S(x))$, where $P,Q,R,S$ are quadratic trinomials.
[i]A. Golovanov[/i]
[b]EDIT.[/b] It is confirmed that assuming the coefficients to be [b]real[/b], while solving the problem, earned a maximum score.
Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.
Cut a square into eight acute-angled triangles.
Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$.
Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$.
[i]Oleksii Masalitin, Ukraine[/i]
Consider the Harmonic Table
\[\begin{array}{c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c}&&&1&&&\\&&\tfrac12&&\tfrac12&&\\&\tfrac13&&\tfrac16&&\tfrac13&\\\tfrac14&&\tfrac1{12}&&\tfrac1{12}&&\tfrac14\\&&&\vdots&&&\end{array}\] where $a_{n,1}=1/n$ and \[a_{n,k+1}=a_{n-1,k}-a_{n,k}.\] Find the remainder when the sum of the reciprocals of the $2007$ terms on the $2007^\text{th}$ row gets divided by $2008$.
A sequence $a_0, a_1, a_2, \dots$ of real numbers satisfies $a_0 = 999$, $a_1 = -999$, and $a_n = a_{n-1}a_{n+1}$ for each positive integer $n$. Compute $\left\lvert a_1 + a_2 + \dots + a_{1000} \right\rvert$.
[i]Proposed by Jeremy Lu[/i]
Find the sum of all integers from 1 to 1000 which contain at least one “7” in their digits.
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.
Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.
(a) Let $x$ be a real number with $x \ge 1$. Prove that $x^3 - 5x^2 + 8x - 4 \ge 0$.
(b) Let $a, b \ge 1$ real numbers. Find the minimum value of the expression $ab(a + b - 10) + 8(a + b)$. Determine
also the real number pairs $(a, b)$ that make this expression equal to this minimum value.
Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$. Let $I$ be the incenter of$ ABC.$
[i](a)[/i] Construct the points $P$ and $Q$ on the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$, where $k$ is a given rational number.
[i](b) [/i]Let $R$ be the intersection point of $PQ$ and $AD$. For what value of $k$ does the equality $AR = RI$ hold?
[i](c)[/i] In which case do the equalities $AR = RI = ID$ hold?