Found problems: 85335
1986 Dutch Mathematical Olympiad, 3
The following apply: $a,b,c,d \ge 0$ and $abcd=1$
Prove that $$ a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd \ge 10$$
LMT Speed Rounds, 5
Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution][i]Solution[/i]. $\boxed{186}$
We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]
2018 Taiwan TST Round 3, 2
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
2010 Moldova Team Selection Test, 1
Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.
2007 Tournament Of Towns, 1
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
2022 Kazakhstan National Olympiad, 1
Given a triangle $ABC$ draw the altitudes $AD$, $BE$, $CF$. Take points $P$ and $Q$ on $AB$ and $AC$, respectively such that $PQ \parallel BC$. Draw the circles with diameters $BQ$ and $CP$ and let them intersect at points $R$
and $T$ where $R$ is closer to $A$ than $T$. Draw the altitudes $BN$ and $CM$ in the triangle $BCR$. Prove that $FM$, $EN$ and $AD$ are concurrent.\\
2005 Indonesia MO, 6
Find all triples $ (x,y,z)$ of integers which satisfy
$ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$
$ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$
$ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.
2016 IFYM, Sozopol, 5
A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides.
a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle.
b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic.
2017 Junior Regional Olympiad - FBH, 4
Let $n$ and $k$ be positive integers for which we have $4$ statements:
$i)$ $n+1$ is divisible with $k$
$ii)$ $n=2k+5$
$iii)$ $n+k$ is divisible with $3$
$iv)$ $n+7k$ is prime
Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false
Ukrainian TYM Qualifying - geometry, 2018.18
In the acute triangle $ABC$, the altitude $AH$ is drawn. Using segments $AB,BH,CH$ and $AC$ as diameters circles $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are constructed respectively. Besides the point $H$, the circles $\omega_1$ and $\omega_3$ intersect at the point $P,$ and the circles $\omega_2$ and $\omega_4$ interext at point $Q$. The lines $BQ$ and $CP$ intersect at point $N$. Prove that this point lies on the midline of triangle $ABC$, which is parallel to $BC$.
2013 Brazil National Olympiad, 5
Let $x$ be an irrational number between 0 and 1 and $x = 0.a_1a_2a_3\cdots$ its decimal representation. For each $k \ge 1$, let $p(k)$ denote the number of distinct sequences $a_{j+1} a_{j+2} \cdots a_{j+k}$ of $k$ consecutive digits in the decimal representation of $x$. Prove that $p(k) \ge k+1$ for every positive integer $k$.
1943 Eotvos Mathematical Competition, 3
Let $a < b < c < d$ be real numbers and $(x,y, z,t)$ be any permutation of $a$,$b$, $c$ and $d$. What are the maximum and minimum values of the expression $$(x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2?$$
2007 Belarusian National Olympiad, 4
Each point of a circle is painted in one of the $ N$ colors ($N \geq 2$). Prove that there exists an inscribed trapezoid such that all its vertices are painted the same color.
2015 ASDAN Math Tournament, 16
Find the maximum value of $c$ such that
\begin{align*}
1&=-cx+y\\
-7&=x^2+y^2+8y
\end{align*}
has a unique real solution $(x,y)$.
2012 USAMTS Problems, 5
An ordered quadruple $(y_1,y_2,y_3,y_4)$ is $\textbf{quadratic}$ if there exist real numbers $a$, $b$, and $c$ such that \[y_n=an^2+bn+c\] for $n=1,2,3,4$.
Prove that if $16$ numbers are placed in a $4\times 4$ grid such that all four rows are quadratic and the first three columns are also quadratic then the fourth column must also be quadratic.
[i](We say that a row is quadratic if its entries, in order, are quadratic. We say the same for a column.)[/i]
[asy]
size(100);
defaultpen(linewidth(0.8));
for(int i=0;i<=4;i=i+1)
draw((i,0)--(i,4));
for(int i=0;i<=4;i=i+1)
draw((0,i)--(4,i));
[/asy]
2004 Alexandru Myller, 4
Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $
[i]Mihai Piticari[/i] and [i]Dan Popescu[/i]
Russian TST 2017, P3
Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.
2016 AMC 10, 11
What is the area of the shaded region of the given $8 \times 5$ rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));
label("$1$",(1/2,5),dir(90));
label("$7$",(9/2,5),dir(90));
label("$1$",(8,1/2),dir(0));
label("$4$",(8,3),dir(0));
label("$1$",(15/2,0),dir(270));
label("$7$",(7/2,0),dir(270));
label("$1$",(0,9/2),dir(180));
label("$4$",(0,2),dir(180));
[/asy]
$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$
2010 LMT, 8
How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$
MathLinks Contest 3rd, 2
Let $k \ge 1$ be an integer and $a_1, a_2, ... , a_k, b1, b_2, ..., b_k$ rational numbers with the property that for any irrational numbers $x_i >1$, $i = 1, 2, ..., k$, there exist the positive integers $n_1, n_2, ... , n_k, m_1, m_2, ..., m_k$ such that $$a_1\lfloor x^{n_1}_1\rfloor + a_2 \lfloor x^{n_2}_2\rfloor + ...+ a_k\lfloor x^{n_k}_k\rfloor=b_1\lfloor x^{m_1}_1\rfloor +2_1\lfloor x^{m_2}_2\rfloor+...+b_k\lfloor x^{m_k}_k\rfloor $$
Prove that $a_i = b_i$ for all $i = 1, 2, ... , k$.
2011 Federal Competition For Advanced Students, Part 2, 2
Let $k$ and $n$ be positive integers.
Show that if $x_j$ ($1\leqslant j\leqslant n$) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$, then
\[\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}\]
2013 AMC 10, 8
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\]
$ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $
2021 Harvard-MIT Mathematics Tournament., 2
Compute the number of ordered pairs of integers $(a, b),$ with $2 \le a, b \le 2021,$ that satisfy the equation
\[a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.\]
2012 India PRMO, 17
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
2014 Bosnia And Herzegovina - Regional Olympiad, 1
Find all possible values of $$\frac{(a+b-c)^2}{(a-c)(b-c)}+\frac{(b+c-a)^2}{(b-a)(c-a)}+\frac{(c+a-b)^2}{(c-b)(a-b)}$$