Found problems: 85335
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
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
1961 Czech and Slovak Olympiad III A, 4
Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.
2003 District Olympiad, 2
Find all functions $\displaystyle f : \mathbb N^\ast \to M$ such that
\[ \displaystyle 1 + f(n) f(n+1) = 2 n^2 \left( f(n+1) - f(n) \right), \, \forall n \in \mathbb N^\ast , \]
in each of the following situations:
(a) $\displaystyle M = \mathbb N$;
(b) $\displaystyle M = \mathbb Q$.
[i]Dinu Şerbănescu[/i]
2013 Kyiv Mathematical Festival, 4
Elza draws $2013$ cities on the map and connects some of them with $N$ roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest $N$ for which Elza has a winning strategy.
1985 All Soviet Union Mathematical Olympiad, 413
Given right hexagon. The lines parallel to all the sides are drawn from all the vertices and midpoints of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto $24$ triangles, and the figure has $19$ nodes. $19$ different numbers are written in those nodes. Prove that at least $7$ of $24$ triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise.
2012 239 Open Mathematical Olympiad, 8
We call a tetrahedron divisor of a parallelepiped if the parallelepiped can be divided into $6$ copies of that tetrahedron. Does there exist a parallelepiped that it has at least two different divisor tetrahedrons?