Found problems: 85335
2006 Princeton University Math Competition, 4
What are the last two digits of $$2003^{{2005}^{{2007}^{2009}}}$$ , where $a^{b{^c}}$ means $a^{(b^c)}$?
2019 Baltic Way, 9
For a positive integer $n$, consider all nonincreasing functions $f : \{1,\hdots,n\}\to\{1,\hdots,n\}$. Some of them have a fixed point (i.e. a $c$ such that $f(c) = c$), some do not. Determine the difference between the sizes of the two sets of functions.
[i]Remark.[/i] A function $f$ is [i]nonincreasing[/i] if $f(x) \geq f(y)$ holds for all $x \leq y$
2015 Indonesia MO Shortlist, A2
Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that
$\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$.
Show that $a$ is a natural number.
LMT Team Rounds 2010-20, 2020.S1
Compute the smallest nonnegative integer that can be written as the sum of 2020 distinct integers.
1978 All Soviet Union Mathematical Olympiad, 267
Given $a_1, a_2, ... , a_n$. Define $$b_k = \frac{a_1 + a_2 + ... + a_k}{k}$$ for $1 \le k\le n.$ Let $$C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2$$
Prove that $C \le D \le 2C$.
2010 Peru MO (ONEM), 3
Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.
2020 AMC 10, 21
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$
$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
2019 Caucasus Mathematical Olympiad, 7
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible
to have equal numbers of apricots in all the boxes after $k$ moves.
2008 May Olympiad, 3
On a blackboard are written all the integers from $1$ to $2008$ inclusive. Two numbers are deleted and their difference is written. For example, if you erase $5$ and $241$, you write $236$. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be $2008$. What about $2007$? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.
PEN E Problems, 27
Prove that for each positive integer $n$, there exist $n$ consecutive positive integers none of which is an integral power of a prime number.
1957 Moscow Mathematical Olympiad, 352
Of all parallelograms of a given area find the one with the shortest possible longer diagonal.
2014 IMS, 1
Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.
2010 Malaysia National Olympiad, 4
In the diagram, $\angle AOB = \angle BOC$ and$\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$. Find $\angle AOF$.
[img]https://cdn.artofproblemsolving.com/attachments/b/2/deba6cd740adbf033ad884fff8e13cd21d9c5a.png[/img]
2018 Brazil Undergrad MO, 2
Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?
2007 Oral Moscow Geometry Olympiad, 6
A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point.
(A. Zaslavsky)
1977 AMC 12/AHSME, 28
Let $g(x)=x^5+x^4+x^3+x^2+x+1$. What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$?
$\textbf{(A) }6\qquad\textbf{(B) }5-x\qquad\textbf{(C) }4-x+x^2\qquad$
$\textbf{(D) }3-x+x^2-x^3\qquad \textbf{(E) }2-x+x^2-x^3+x^4$
2015 Mathematical Talent Reward Programme, MCQ: P 13
Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable?
[list=1]
[*] 0
[*] 2
[*] 4
[*] $\infty$
[/list]
2016 NIMO Problems, 8
Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x, y,$ or $z$-directions. However, the evil David anticipated Justin's arrival, and so he has surreptitiously placed a mine at the point $(2,2,2)$. If at any point Justin is at most one unit away from this mine (in any direction), the mine detects his presence and explodes, thwarting Justin.
How many paths can Justin take to reach his destination safely?
[i]Proposed by Justin Stevens[/i]
2018 Brazil Team Selection Test, 5
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.
2019 Silk Road, 2
Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $
Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .
2025 Malaysian APMO Camp Selection Test, 2
There are $n\ge 3$ students in a classroom. Every day, the teacher separates the students into at least two non-empty groups, and each pair of students from the same group will shake hands once. Suppose after $k$ days, each pair of students has shaken hands exactly once, and $k$ is as minimal as possible. Prove that $$\sqrt{n} \le k-1 \le 2\sqrt{n}$$
[i]Proposed by Wong Jer Ren[/i]
2019 Poland - Second Round, 3
Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and:
\begin{align*}
\underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)).
\end{align*}
2003 Baltic Way, 18
Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$?
[size=85][color=#0000FF][Mod edit: Question fixed][/color][/size]
1992 IMO Longlists, 81
Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$
2001 Romania National Olympiad, 1
Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.