Found problems: 85335
2009 Romania National Olympiad, 1
Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by
\[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\]
and $x_0,y_0$ are given real numbers.
a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit.
b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.
2007 Postal Coaching, 5
There are $N$ points in the plane such that the [b]total number[/b] of pairwise distances of these $N$ points is at most $n$. Prove that $N \le (n + 1)^2$.
2016 PUMaC Algebra Individual B, B1
If $x$ is a positive real number such that $(x^2 - 1)^2 - 1 = 9800$, compute $x$.
1959 Poland - Second Round, 2
What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?
2023 HMNT, 32
Compute $$\sum_{\underset{a \ge 6, b,c \ge 0}{a+b+c=12}} \frac{a!}{b!c!(a-b-c)!},$$ where the sum runs over all triples of nonnegative integers $(a,b,c)$ such that $a+b+c=12$ and $a \ge 6.$
2016 Tournament Of Towns, 4
A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.)
[i](8 points)[/i]
[i]Mikhail Evdokimov[/i]
1989 Chile National Olympiad, 1
Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$
1969 AMC 12/AHSME, 8
Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x+75^\circ$, $2x+25^\circ$, $3x-22^\circ$. Then one interior angle of the triangle, in degrees, is:
$\textbf{(A) }57\tfrac12\qquad
\textbf{(B) }59\qquad
\textbf{(C) }60\qquad
\textbf{(D) }61\qquad
\textbf{(E) }122$
2004 USA Team Selection Test, 3
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
2017 AIME Problems, 4
A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
2014 Greece Team Selection Test, 3
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
1999 Gauss, 20
The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$.
\[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \\ \hline 5 & C & 13\\ \hline D & E & 3 \\ \hline\end{tabular} \]
$\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 16$
1983 AMC 12/AHSME, 15
Three balls marked 1,2, and 3, are placed in an urn. One ball is drawn, its number is recorded, then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is 6, what is the probability that the ball numbered 2 was drawn all three times?
$\displaystyle \text{(A)} \ \frac{1}{27} \qquad \text{(B)} \ \frac{1}{8} \qquad \text{(C)} \ \frac{1}{7} \qquad \text{(D)} \ \frac{1}{6} \qquad \text{(E)} \ \frac{1}{3}$
2022 IFYM, Sozopol, 6
For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that
$$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$
$$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$
Prove that for every natural number $n$ the following inequality holds:
$$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$
2012 District Olympiad, 2
If $ a,b,c>0, $ then $ \sum_{\text{cyc}} \frac{a}{2a+b+c}\le 3/4. $
2025 CMIMC Combo/CS, 4
Let $n$ and $k$ be positive integers, with $k \le n.$ Define a (simple, undirected) graph $G_{n,k}$ as follows: its vertices are all of the binary strings of length $n,$ and there is an edge between two strings if and only if they differ in exactly $k$ positions. If $c_{n,k}$ denotes the number of connected components of $G_{n,k},$ compute $$\sum_{n=1}^{10} \sum_{k=1}^n c_{n,k}.$$ (For example, $G_{3,2}$ has two connected components.)
2023 HMNT, 1
Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler’s series.
2015 Baltic Way, 20
For any integer $n \ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \ge 2 $ for which $ A_n$ is odd. (Note: The distance between two integers $ a$ and $b$ is defined as $|a -b|$.)
2017 Korea Winter Program Practice Test, 4
For a nonzero integer $k$, denote by $\nu_2(k)$ the maximal nonnegative integer $t$ such that $2^t \mid k$. Given are $n (\ge 2)$ pairwise distinct integers $a_1, a_2, \ldots, a_n$. Show that there exists an integer $x$, distinct from $a_1, \ldots, a_n$, such that among $\nu_2(x - a_1), \ldots, \nu_2(x - a_n)$ there are at least $n/4$ odd numbers and at least $n/4$ even numbers.
2017 Novosibirsk Oral Olympiad in Geometry, 2
You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.
2018 CMIMC Individual Finals, 2
Compute the sum of the digits of \[\prod_{n=0}^{2018}\left(10^{2\cdot 3^n} - 10^{3^n} + 1\right)\left(10^{2\cdot 3^n} + 10^{3^n} + 1\right).\]
2021 Philippine MO, 7
Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and
$$a+b+c+d = 13$$
$$a^2+b^2+c^2+d^2=43.$$
Show that $ab \geq 3 + cd$.
1994 Tournament Of Towns, (408) 6
At each integer point of the numerical line a lamp with a toggle button is placed. If the button is pressed, a lit lamp is turned off, an unlit one is turned on. Initially all the lamps are unlit. A stencil with a finite set of fixed holes at integer distances is chosen. The stencil may be moved along the line as a rigid body, and for any fixed position of the stencil, one may push simultaneously all the buttons accessible through the holes. Prove that for any stencil it is possible to get exactly two lit lamps after several such operations.
(B Ginsburg)
2018 China Team Selection Test, 6
Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .
2005 France Team Selection Test, 5
Let $ABC$ be a triangle such that $BC=AC+\frac{1}{2}AB$. Let $P$ be a point of $AB$ such that $AP=3PB$.
Show that $\widehat{PAC} = 2 \widehat{CPA}.$