Found problems: 85335
2006 Pan African, 4
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.
2017 Romania National Olympiad, 3
Let be two natural numbers $ n $ and $ a. $
[b]a)[/b] Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality.
$$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$
[b]b)[/b] Show that there exist only finitely such $ n\text{-tuplets} . $
2018 Ramnicean Hope, 1
Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $
[i]Mihai Neagu[/i]
2018 Czech and Slovak Olympiad III A, 1
In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. [i](Since this was the 67th year of CZE/SVK MO,)[/i] show that if the group is 6-great, then it is 7-great as well.
[b]Bonus[/b] (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is $(k+1)$-great as well.
2011 Greece Junior Math Olympiad, 2
We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.
2009 Jozsef Wildt International Math Competition, W. 18
If $a$, $b$, $c>0$ and $abc=1$, then $$\sum \limits^{cyc} \frac{a+b+c^n}{a^{2n+3}+b^{2n+3}+ab} \leq a^{n+1}+b^{n+1}+c^{n+1}$$ for all $n\in \mathbb{N}$
1983 Swedish Mathematical Competition, 1
The positive integers are grouped as follows: $1, 2+3, 4+5+6, 7+8+9+10,\dots$. Find the value of the $n$-th sum.
2009 Today's Calculation Of Integral, 457
Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$
2024 Moldova EGMO TST, 8
In the plane there are $n$ $(n\geq4)$ marked points. There are at least $n+1$ pairs of marked points such that the distance between each pair of points is $1$. Find the greatest integer $k$ such that there is a marked point that is the center of the circle with radius $1$ on which there are at least $k$ of the marked points.
2020 Malaysia IMONST 1, 11
If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest
possible value of $p$.
2024-25 IOQM India, 26
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
2025 Vietnam National Olympiad, 1
Let $P(x) = x^4-x^3+x$.
a) Prove that for all positive real numbers $a$, the polynomial $P(x) - a$ has a unique positive zero.
b) A sequence $(a_n)$ is defined by $a_1 = \dfrac{1}{3}$ and for all $n \geq 1$, $a_{n+1}$ is the positive zero of the polynomial $P(x) - a_n$. Prove that the sequence $(a_n)$ converges, and find the limit of the sequence.
1991 IMTS, 5
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?
2010 China Girls Math Olympiad, 7
For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.
1987 Canada National Olympiad, 3
Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?
1999 Mongolian Mathematical Olympiad, Problem 4
Investigate if there exist infinitely many natural numbers $n$ such that $n$ divides $2^n+3^n$.
2003 Junior Tuymaada Olympiad, 4
The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$
2025 Ukraine National Mathematical Olympiad, 10.7
In a row, $1000$ numbers \(2\) and $2000$ numbers \(-1\) are written in some order.
Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals \(0\).
(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.
[i]Proposed by Anton Trygub[/i]
2009 Portugal MO, 2
Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$, respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$.
1997 IMO, 3
Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions:
\[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\
|x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right.
\]
Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that
\[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}.
\]
2019 Math Prize for Girls Problems, 16
The figure shows a regular heptagon with sides of length 1.
[asy]
import geometry;
unitsize(5);
real R = 1/(2 sin(pi/7));
pair A = (0, R);
pair B = rotate(360/7) * A;
pair C = rotate(360/7) * B;
pair D = rotate(360/7) * C;
pair E = rotate(360/7) * D;
pair F = rotate(360/7) * E;
pair G = rotate(360/7) * F;
pair X = B + G - A;
pair Y = (D + E) / 2;
draw(A -- B -- C -- D -- E -- F -- G -- cycle);
draw("$1$", B -- X);
draw("$1$", X -- G);
draw("$d$", X -- Y);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(X);
dot(Y);
perpendicular(Y, NW, Y - A);
[/asy]
Determine the indicated length $d$. Express your answer in simplified radical form.
2021 Science ON grade VIII, 3
$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that
$$\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).$$
[i] (Adapted from folklore)[/i]
2023 4th Memorial "Aleksandar Blazhevski-Cane", P6
Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
[b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$.
[b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$.
[i]Proposed by Nikola Velov[/i]
2020 Baltic Way, 16
Richard and Kaarel are taking turns to choose numbers from the set $\{1,\dots,p-1\}$ where $p > 3$ is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel.
Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by $p$. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by $p$. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?
Brazil L2 Finals (OBM) - geometry, 2011.2
Let $ ABCD $ be a convex quadrilateral such that $ AD = DC, AC = AB $ and $ \angle ADC = \angle CAB $. If $ M $ and $ N $ are midpoints of the $ AD $ and $ AB $ sides, prove that the $ MNC $ triangle is isosceles.