Found problems: 85335
2002 Estonia National Olympiad, 4
Find the maximum length of a broken line on the surface of a unit cube, such that its links are the cube’s edges and diagonals of faces, the line does not intersect itself and passes no more than once through any vertex of the cube, and its endpoints are in two opposite vertices of the cube.
1996 Argentina National Olympiad, 5
Determine all positive real numbers $x$ for which $$\left [x\right ]+\left [\sqrt{1996x}\right ]=1996$$ is verified
Clarification:The brackets indicate the integer part of the number they enclose.
2024 AMC 10, 7
What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }7 \qquad
\textbf{(D) }11 \qquad
\textbf{(E) }18 \qquad
$
1991 Mexico National Olympiad, 4
The diagonals $AC$ and $BD$ of a convex quarilateral $ABCD$ are orthogonal. Let $M,N,R,S$ be the midpoints of the sides $AB,BC,CD$ and $DA$ respectively, and let $W,X,Y,Z$ be the projections of the points $M,N,R$ and $S$ on the lines $CD,DA,AB$ and $BC$, respectively. Prove that the points $M,N,R,S,W,X,Y$ and $Z$ lie on a circle.
2024 Bulgarian Autumn Math Competition, 10.3
Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.
2019 USAMTS Problems, 3
A positive integer $n > 1$ is juicy if its divisors $d_1 < d_2 < \dots < d_k$ satisfy $d_i - d_{i-1} \mid n$ for all $2 \leq i \leq k$. Find all squarefree juicy integers.
2025 Belarusian National Olympiad, 10.4
Is it possible to assign every integral point $(x,y)$ of the plane a positive integer $a_{x,y}$ such that for every two integers $i$ and $j$ the following equality holds $$a_{i,j}=\gcd(a_{i-1,j},a_{i+1,j})+\gcd(a_{i,j-1},a_{i,j+1})$$
[i]M. Shutro[/i]
2022 Flanders Math Olympiad, 2
A domino is a rectangle whose length is twice its width.
Any square can be divided into seven dominoes, for example as shown in the figure below.
[img]https://cdn.artofproblemsolving.com/attachments/7/6/c055d8d2f6b7c24d38ded7305446721e193203.png[/img]
a) Show that you can divide a square into $n$ dominoes for all $n \ge 5$.
b) Show that you cannot divide a square into three or four dominoes.
2009 AIME Problems, 6
How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} \equal{} N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.)
2023-IMOC, N5
Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.
2005 India National Olympiad, 3
Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that
$a)$ $\alpha$ is real and negative;
$b)$ the remaining third quadratic equation has non-real roots.
2012 Estonia Team Selection Test, 4
Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.
PEN A Problems, 113
Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]
2016 Brazil Undergrad MO, 6
Let it \(C,D > 0\). We call a function \(f:\mathbb{R} \rightarrow \mathbb{R}\) [i]pretty[/i] if \(f\) is a \(C^2\)-class, \(|x^3f(x)| \leq C\) and \(|xf''(x)| \leq D\).
[list='i']
[*] Show that if \(f\) is pretty, then, given \(\epsilon \geq 0\), there is a \(x_0 \geq 0\) such that for every \(x\) with \(|x| \geq x_0\), we have \(|x^2f'(x)| < \sqrt{2CD}+\epsilon\).
[*] Show that if \(0 < E < \sqrt{2CD}\) then there is a pretty function \(f\) such that for every \(x_0 \geq 0\) there is a \(x > x_0\) such that \(|x^2f'(x)| > E\).
[/list]
2010 Contests, 2
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$
2016 India IMO Training Camp, 1
An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]
2014 Contests, 2
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
2010 Regional Olympiad of Mexico Center Zone, 2
Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.
1989 Swedish Mathematical Competition, 5
Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$. Prove that if $a > 0$, then
$$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$
2020 Israel National Olympiad, 5
Two triangles $ACE, BDF$ are given which intersect at six points: $G, H, I, J, K, L$ as in the picture. It is known that in each of the quadrilaterals
\[ABIK ,BCJL ,CDKG ,DELH ,EFGI\]
it is possible to inscribe a circle. Is it possible for the quadrilateral $FAHJ$ is also circumscribed around a circle?
2019 India PRMO, 10
Let $ABC$ be a triangle and let $\Omega$ be its circumcircle. The internal bisectors of angles $A, B$ and $C$ intersect $\Omega$ at $A_1, B_1$ and $C_1$, respectively, and the internal bisectors of angles $A_1, B_1$ and $C_1$ of the triangles $A_1 A_2 A_ 3$ intersect $\Omega$ at $A_2, B_2$ and $C_2$, respectively. If the smallest angle of the triangle $ABC$ is $40^{\circ}$, what is the magnitude of the smallest angle of the triangle $A_2 B_2 C_2$ in degrees?
2014 Postal Coaching, 4
Given arbitrary complex numbers $w_1,w_2,\ldots,w_n$, show that there exists a positive integer $k\le 2n+1$ for which $\text{Re} (w_1^k+w_2^k+\cdots+w_n^k)\ge 0$.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, A1
Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$
2003 National Olympiad First Round, 24
If $3a=1+\sqrt 2$, what is the largest integer not exceeding $9a^4-6a^3+8a^2-6a+9$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1969 Polish MO Finals, 2
Given distinct real numbers $a_1,a_2,...,a_n$, find the minimum value of the function
$$y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.$$