Found problems: 85335
2004 Baltic Way, 3
Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]
Oliforum Contest IV 2013, 7
For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.
2014 BMT Spring, 4
What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$?
2023 Girls in Mathematics Tournament, 4
Determine all $n$ positive integers such that exists an $n\times n$ where we can write $n$ times each of the numbers from $1$ to $n$ (one number in each cell), such that the $n$ sums of numbers in each line leave $n$ distinct remainders in the division by $n$, and the $n$ sums of numbers in each column leave $n$ distinct remainders in the division by $n$.
2008 Harvard-MIT Mathematics Tournament, 2
Find the smallest positive integer $ n$ such that $ 107n$ has the same last two digits as $ n$.
2019 AIME Problems, 1
Consider the integer $$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.$$ Find the sum of the digits of $N$.
2021 Tuymaada Olympiad, 2
In trapezoid $ABCD$,$M$ is the midpoint of the base $AD$.Point $E$ lies on the segment $BM$.It is known that $\angle ADB=\angle MAE=\angle BMC$.Prove that the triangle $BCE $ is isosceles.
2013 Spain Mathematical Olympiad, 4
Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.
2022 Brazil National Olympiad, 4
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?
1998 Federal Competition For Advanced Students, Part 2, 3
Let $a_n$ be a sequence recursively de fined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$. For what value of the base $b$ we get the sum $1$?
2009 IberoAmerican Olympiad For University Students, 4
Given two positive integers $m,n$, we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties:
i) $f$ is continuous;
ii) $f(0) = 0$, $f(m) = n$;
iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$, then $t_2-t_1 \in \{0,m\}$.
Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$-slippery.
2007 Today's Calculation Of Integral, 179
Evaluate the following integrals.
(1) Meiji University
$\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$
(2) Tokyo University of Science
$\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$
2024 Belarus Team Selection Test, 3.1
Triangles $ABC$ and $DEF$, having a common incircle of radius $R$, intersect at points
$X_1, X_2, \ldots , X_6$ and form six triangles (see the figure below). Let $r_1, r_2,\ldots, r_6$ be the radii of the
inscribed circles of these triangles, and let $R_1, R_2, \ldots , R_6$ be the radii of the inscribed circles of the
triangles $AX_1F, FX_2B, BX_3D, DX_4C, CX_5E$ and $EX_6A$ respectively.
[img]https://i.ibb.co/BspgdHB/Image.jpg[/img]
Prove that \[ \sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i} \]
[i]U. Maksimenkau[/i]
2008 Oral Moscow Geometry Olympiad, 6
Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$.
(A. Zaslavsky)
2008 All-Russian Olympiad, 5
Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]
2011 Junior Balkan Team Selection Tests - Romania, 3
a) Find the largest possible value of the number $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$, if $x_1, x_2, ... , x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$.
b) Find the numbers $x_1, x_2, ... , x_n$ for which the maximum value determined at a) is obtained
2015 India IMO Training Camp, 3
Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.
1953 Putnam, A6
Show that the sequence
$$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$
converges and evaluate the limit.
1992 AIME Problems, 9
Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2021 Regional Olympiad of Mexico West, 6
Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles.
Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.
1990 USAMO, 2
A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows: \begin{align*}f_1(x) &= \sqrt{x^2 + 48}, \quad \mbox{and} \\ f_{n+1}(x) &= \sqrt{x^2 + 6f_n(x)} \quad \mbox{for } n \geq 1.\end{align*}
(Recall that $\sqrt{\makebox[5mm]{}}$ is understood to represent the positive square root.) For each positive integer $n$, find all real solutions of the equation $\, f_n(x) = 2x \,$.
2023 Dutch BxMO TST, 5
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
2005 Tournament of Towns, 4
For any function $f(x)$, define $f^1(x) = f(x)$ and $f^n (x) = f(f^{n-1}(x))$ for any integer $n \ge 2$. Does there exist a quadratic polynomial $f(x)$ such that the equation $f^n(x) = 0$ has exactly $2^n$ distinct real roots for every positive integer $n$?
[i](6 points)[/i]
2011 Argentina National Olympiad, 3
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 75^o$ and $AB = 2$. The points $P$ and $Q$ on the sides $AC$ and $BC$ respectively are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$ . Calculate the measurement of the segment $QA $.
2017 China Team Selection Test, 3
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$