Found problems: 85335
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
$$y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.$$
2007 Mongolian Mathematical Olympiad, Problem 2
Given $101$ segments in a line, prove that there exists $11$ segments meeting in $1$ point or $11$ segments such that every two of them are disjoint.