Found problems: 85335
1983 IMO Longlists, 41
Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?
2017 Australian MO, 4
Find all pairs $(a,b)$ of non-negative integers such that $2017^a=b^6-32b+1$.
1978 Chisinau City MO, 156
The natural numbers $a_1 <a_2 <.... <a_n\le 2n$ are such that the least common multiple of any two of them is greater than $2n$. Prove that $a_1 >\left[\frac{2n}{3}\right]$.
2006 Princeton University Math Competition, 2
Express $\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}$ in the simplest possible form.
2021 Turkey Junior National Olympiad, 1
Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.
2012 Stars of Mathematics, 3
For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality
$$ \left | \dfrac {a} {b-c} \right | + \left | \dfrac {b} {c-a} \right | + \left | \dfrac {c} {a-b} \right | \geq 2$$
and determine all cases of equality.
Prove that if we also impose $a,b,c$ positive, then all equality cases disappear, but the value $2$ remains the best constant possible.
([i]Dan Schwarz[/i])
2023 IFYM, Sozopol, 4
Let $n$ be a natural number. The leader of the math team invites $n$ girls for winter training, and each leaves her two gloves in a common box upon entry. The mischievous little brother randomly pairs the gloves into pairs, where each pair consists of one left glove and one right glove. A pairing is called [i]weak[/i] if there is a set of $k < \frac{n}{2}$ pairs containing gloves of exactly $k$ girls. Find the probability that the pairing is not weak.
2003 Finnish National High School Mathematics Competition, 1
The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2020 LMT Fall, A26
Jeff has planted $7$ radishes, labelled $R$, $A$, $D$, $I$, $S$, $H$, and $E$. Taiki then draws circles through $S,H,I,E,D$, then through $E,A,R,S$, and then through $H,A,R,D$, and notices that lines drawn through $SH$, $AR$, and $ED$ are parallel, with $SH = ED$. Additionally, $HER$ is equilateral, and $I$ is the midpoint of $AR$. Given that $HD = 2$, $HE$ can be written as $\frac{-\sqrt{a} + \sqrt{b} + \sqrt{1+\sqrt{c}}}{2}$, where $a,b,$ and $c$ are integers, find $a+b+c$.
[i]Proposed by Jeff Lin[/i]
1993 Iran MO (2nd round), 3
Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that
\[ 1 < \frac ab < 1 +\frac 1n.\]
Durer Math Competition CD Finals - geometry, 2008.D1
Given a square grid where the distance between two adjacent grid points is $1$. Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?
1956 AMC 12/AHSME, 9
Simplify $ \left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$; the result is:
$ \textbf{(A)}\ a^{16} \qquad\textbf{(B)}\ a^{12} \qquad\textbf{(C)}\ a^8 \qquad\textbf{(D)}\ a^4 \qquad\textbf{(E)}\ a^2$
2007 Turkey Team Selection Test, 3
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]
2002 Romania National Olympiad, 1
Let $A$ be a ring.
$a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$.
$b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.
2007 Tuymaada Olympiad, 3
$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are altitudes of an acute triangle $ ABC$. A circle passing through $ A_{1}$ and $ B_{1}$ touches the arc $ AB$ of its circumcircle at $ C_{2}$. The points $ A_{2}$, $ B_{2}$ are defined similarly. Prove that the lines $ AA_{2}$, $ BB_{2}$, $ CC_{2}$ are concurrent.
2012 Greece JBMO TST, 1
Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$
2020 ELMO Problems, P3
Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.
[i]Proposed by Fedir Yudin.[/i]
2000 Harvard-MIT Mathematics Tournament, 2
Simplify $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^6+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^6$ to the form $a+bi$.
2023 India IMO Training Camp, 2
Let $\mathbb R^+$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ satisfying \[f(x+y^2f(x^2))=f(xy)^2+f(x)\] for all $x,y \in \mathbb{R}^+$.
[i]Proposed by Shantanu Nene[/i]
1997 IMO Shortlist, 6
(a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that
\[ x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}.\]
(b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that
\[ x^a \plus{} y^b \equal{} z^c.\]
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$
2007 AMC 10, 7
Last year Mr. John Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he has left in state taxes. He paid a total of $ \$10,500$ for both taxes. How many dollars was the inheritance?
$ \textbf{(A)}\ 30,000 \qquad \textbf{(B)}\ 32,500 \qquad \textbf{(C)}\ 35,000 \qquad \textbf{(D)}\ 37,500 \qquad \textbf{(E)}\ 40,000$
2015 IFYM, Sozopol, 1
Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.
2015 QEDMO 14th, 1
Let $n$ be a natural number. A regular hexagon with edge length $n$ gets split into equilateral exploded triangles whose edges are $1$ in length and parallel to one side of the hexagon. Find the number of regular hexagons, the angles of which are all angles of these triangles are.
2024 UMD Math Competition Part II, #5
Define two sequences $x_n, y_n$ for $n = 1, 2, \ldots$ by \[x_n = \left(\sum^n_{k=0} \binom{2n}{2k}49^k 48^{n-k} \right) -1, \quad \text{and} \quad y_n = \sum^{n-1}_{k=0} \binom{2n}{2k + 1} 49^k 48^{n-k}\] Prove there is a positive integer $m$ for which for every integer $n > m,$ the greatest common factor of $x_n$ and $y_n$ is more than $10^{2024}.$