Found problems: 85335
1978 Bundeswettbewerb Mathematik, 3
For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$
2015 Polish MO Finals, 3
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.
2008 IberoAmerican Olympiad For University Students, 6
[i][b]a)[/b][/i] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^2+B^2=C^2$.
[b][i]b)[/i][/b] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^4+B^4=C^4$.
[b]Note[/b]: The notation $A\in \mathrm{SL}_{2}(\mathbb{Z})$ means that $A$ is a $2\times 2$ matrix with integer entries and $\det A=1$.
1999 National Olympiad First Round, 26
Let $ x$, $ y$, $ z$ be integers such that
\[ \begin{array}{l} {x \minus{} 3y \plus{} 2z \equal{} 1} \\
{2x \plus{} y \minus{} 5z \equal{} 7} \end{array}
\]
Then $ z$ can be
$\textbf{(A)}\ 3^{111} \qquad\textbf{(B)}\ 4^{111} \qquad\textbf{(C)}\ 5^{111} \qquad\textbf{(D)}\ 6^{111} \qquad\textbf{(E)}\ \text{None}$
2008 Purple Comet Problems, 1
Find the greatest prime factor of the sum of the two largest two-digit prime numbers.
2010 Chile National Olympiad, 3
The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.
1955 AMC 12/AHSME, 9
A circle is inscribed in a triangle with sides $ 8$, $ 15$, and $ 17$. The radius of the circle is:
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 7$
2005 MOP Homework, 3
Prove that the equation $a^3-b^3=2004$ does not have any solutions in positive integers.
2010 Today's Calculation Of Integral, 526
For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?
2014 USAMTS Problems, 1:
Fill in each blank unshaded cell with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(9cm);
for (int x=0; x<=11; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (11, y), linewidth(.5pt));
filldraw((0,4)--(0,3)--(2,3)--(2,4)--cycle, gray, gray);
filldraw((1,1)--(1,2)--(3,2)--(3,1)--cycle, gray, gray);
filldraw((4,1)--(4,4)--(5,4)--(5,1)--cycle, gray, gray);
filldraw((7,0)--(7,3)--(6,3)--(6,0)--cycle, gray, gray);
filldraw((7,4)--(7,5)--(6,5)--(6,4)--cycle, gray, gray);
filldraw((8,1)--(8,2)--(10,2)--(10,1)--cycle, gray, gray);
filldraw((9,4)--(9,3)--(11,3)--(11,4)--cycle, gray, gray);
draw((0,0)--(11,0)--(11,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(1, 2, "10");
foo(4, 0, "31");
foo(5, 0, "26");
foo(10, 0, "59");
foo(0, 4, "3");
foo(7, 4, "59");
[/asy]
PEN A Problems, 14
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.
2011 Sharygin Geometry Olympiad, 19
Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?
2014 Baltic Way, 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
2011 Purple Comet Problems, 19
How many ordered pairs of sets $(A, B)$ have the properties:
1. $ A\subseteq \{1, 2, 3, 4, 5, 6\} $
2. $ B\subseteq\{2, 3, 4, 5, 6, 7, 8\} $
3. $ A\cap B $ has exactly $3$ elements.
1949 Moscow Mathematical Olympiad, 160
Prove that for any triangle the circumscribed circle divides the line segment connecting the center of its inscribed circle with the center of one of the exscribed circles in halves.
2008 Harvard-MIT Mathematics Tournament, 5
Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.
1988 Austrian-Polish Competition, 1
Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.
2002 India IMO Training Camp, 20
Let $a,b,c$ be positive real numbers. Prove that
\[\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]
2012 Iran Team Selection Test, 3
The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that
\[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\]
where $S_X$ denotes the surface of figure $X$.
[i]Proposed by Morteza Saghafian, Ali khezeli[/i]
2004 Federal Math Competition of S&M, 2
The sequence $(a_n)$ is determined by $a_1 = 0$ and
$(n+1)^3a_{n+1} = 2n^2(2n+1)a_n+2(3n+1)$ for $n \geq 1$.
Prove that infinitely many terms of the sequence are positive integers.
2008 IberoAmerican, 1
The integers from 1 to $ 2008^2$ are written on each square of a $ 2008 \times 2008$ board. For every row and column the difference between the maximum and minimum numbers is computed. Let $ S$ be the sum of these 4016 numbers. Find the greatest possible value of $ S$.
2003 JHMMC 8, 29
How many three-digit numbers are perfect squares?
2016 Canadian Mathematical Olympiad Qualification, 5
Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.
1977 Chisinau City MO, 135
Solve the equation:
$$x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}$$
1970 AMC 12/AHSME, 3
If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is:
$\textbf{(A) }\dfrac{x+1}{x-1}\qquad\textbf{(B) }\dfrac{x+2}{x-1}\qquad\textbf{(C) }\dfrac{x}{x-1}\qquad\textbf{(D) }2-x\qquad \textbf{(E) }\dfrac{x-1}{x}$