Found problems: 85335
1988 Dutch Mathematical Olympiad, 1
The real numbers $x_1,x_2,..., x_n$ and $a_0,a_1,...,a_{n-1}$ with $x_i \ne 0$ for $i \in\{1,2,.., n\}$ are such that
$$(x-x_1)(x-x_2)...(x-x_n)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$
Express $x_1^{-2}+x_2^{-2}+...+ x_n^{-2}$ in terms of $a_0,a_1,...,a_{n-1}$.
PEN H Problems, 20
Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.
MathLinks Contest 3rd, 2
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles has radius $t$. Prove that
$$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$
2011 Greece Team Selection Test, 3
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
1989 IMO Longlists, 50
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
2021 Dutch Mathematical Olympiad, 3
A frog jumps around on the grid points in the plane, from one grid point to another. The frog starts at the point $(0, 0)$. Then it makes, successively, a jump of one step horizontally, a jump of $2$ steps vertically, a jump of $3$ steps horizontally, a jump of $4$ steps vertically, et cetera. Determine all $n > 0$ such that the frog can be back in $(0, 0)$ after $n$ jumps.
2009 Estonia Team Selection Test, 4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
2004 Germany Team Selection Test, 1
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
1970 Spain Mathematical Olympiad, 5
In the sixth-year exams of a Center, they pass Physics at least$70\%$ of the students, Mathematics at least $75\%$; Philosophy at least, the $90\%$ and the Language at least, $85\%$. How many students, at least, pass these four subjects?
2008 Mathcenter Contest, 5
Let $a,b,c$ be positive real numbers where $ab+bc+ca = 3$. Prove that $$\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\geq\dfrac{3} {2}.$$
[i](dektep)[/i]
2005 MOP Homework, 3
Squares of an $n \times n$ table ($n \ge 3$) are painted black and white as in a chessboard. A move allows one to choose any $2 \times 2$ square and change all of its squares to the opposite color. Find all such n that there is a finite number of the moves described after which all squares are the same color.
2010 District Olympiad, 1
A right that passes through the incircle $ I$ of the triangle $ \Delta ABC$ intersects the side $ AB$ and $ CA$ in $ P$, respective $ Q$. We denote $ BC\equal{}a\ , \ AC\equal{}b\ ,\ AB\equal{}c$ and $ \frac{PB}{PA}\equal{}p\ ,\ \frac{QC}{QA}\equal{}q$.
i) Prove that:
\[ a(1\plus{}p)\cdot \overrightarrow{IP}\equal{}(a\minus{}pb)\overrightarrow{IB}\minus{}pc\overrightarrow{IC}\]
ii) Show that $ a\equal{}bp\plus{}cq$.
iii) If $ a^2\equal{}4bcpq$, then the rights $ AI\ ,\ BQ$ and $ CP$ are concurrents.
1946 Moscow Mathematical Olympiad, 121
Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.
2024 Brazil Cono Sur TST, 3
For a pair of integers $a$ and $b$, with $0<a<b<1000$, a set $S\subset \begin{Bmatrix}1,2,3,...,2024\end{Bmatrix}$ $escapes$ the pair $(a,b)$ if for any elements $s_1,s_2\in S$ we have $\left|s_1-s_2\right| \notin \begin{Bmatrix}a,b\end{Bmatrix}$. Let $f(a,b)$ be the greatest possible number of elements of a set that escapes the pair $(a,b)$. Find the maximum and minimum values of $f$.
2015 BAMO, 3
Which number is larger, $A$ or $B$, where
$$A = \dfrac{1}{2015} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2015})$$
and
$$B = \dfrac{1}{2016} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2016}) \text{ ?}$$
Prove your answer is correct.
2004 Moldova Team Selection Test, 9
Let $a,b$ and $c$ be positive real numbers . Prove that\[\left | \frac{4(b^3-c^3)}{b+c}+ \frac{4(c^3-a^3)}{c+a}+ \frac{4(a^3-b^3)}{a+b} \right |\leq (b-c)^2+(c-a)^2+(a-b)^2.\]
2011 IFYM, Sozopol, 2
prove that $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) \leq 1$ for $0 < a < b \leq c < d$
and when $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) = 1 $
2017 Yasinsky Geometry Olympiad, 2
Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$. The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$.
2009 Indonesia TST, 4
Sixteen people for groups of four people such that each two groups have at most two members in common. Prove that there exists a set of six people in which every group is not properly contained in it.
2021 USAMTS Problems, 2
Find, with proof, the minimum positive integer n with the following property: for
any coloring of the integers $\{1, 2, . . . , n\}$ using the colors red and blue (that is, assigning the
color “red” or “blue” to each integer in the set), there exist distinct integers a, b, c between
1 and n, inclusive, all of the same color, such that $2a + b = c.$
1997 AMC 12/AHSME, 10
Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $ 4$ replaced by $ 3$ and the other die has the $ 3$ replaced by $ 4$. When these dice are rolled, what is the probability that the sum is an odd number?
$ \textbf{(A)}\ \frac{1}{3}\qquad
\textbf{(B)}\ \frac{4}{9}\qquad
\textbf{(C)}\ \frac{1}{2}\qquad
\textbf{(D)}\ \frac{5}{9}\qquad
\textbf{(E)}\ \frac{11}{18}$
2020 China Second Round Olympiad, 4
Given a convex polygon with 20 vertexes, there are many ways of traingulation it (as 18 triangles). We call the diagram of triangulation, meaning the 20 vertexes, with 37 edges(17 triangluation edges and the original 20 edges), a T-diagram. And the subset of this T-diagram with 10 edges which covers all 20 vertexes(meaning any two edges in the subset doesn't cover the same vertex) calls a "perfect matching" of this T-diagram. Among all the T-diagrams, find the maximum number of "perfect matching" of a T-diagram.
2002 AIME Problems, 4
Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5.$
[asy]
size(250);int i,j;
real r=sqrt(3);
for(i=0; i<6; i=i+1) {
for(j=0; j<4; j=j+1) {
draw(shift(((j*r)*dir(60*i+150)).x, ((j*r)*dir(60*i+150)).y)*shift((4r*dir(60i+30)).x,(4r*dir(60i+30)).y)*polygon(6));
}}[/asy]
If $n=202,$ then the area of the garden enclosed by the path, not including the path itself, is $m(\sqrt{3}/2)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000.$
2022 Bulgarian Autumn Math Competition, Problem 8.4
Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence:
$\bullet$ $n_{i+1}\geq n_i$
$\bullet$ There is at least one number $i$, such that $n_i=2022$
$\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$
2003 Baltic Way, 11
Is it possible to select $1000$ points in the plane so that $6000$ pairwise distances between them are equal?