Found problems: 85335
2016 AMC 12/AHSME, 12
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$
2022 Bulgarian Spring Math Competition, Problem 11.3
In every cell of a table with $n$ rows and $m$ columns is written one of the letters $a$, $b$, $c$. Every two rows of the table have the same letter in at most $k\geq 0$ positions and every two columns coincide at most $k$ positions. Find $m$, $n$, $k$ if
\[\frac{2mn+6k}{3(m+n)}\geq k+1\]
2017-IMOC, G6
A point $P$ lies inside $\vartriangle ABC$ such that the values of areas of $\vartriangle PAB, \vartriangle PBC, \vartriangle PCA$ can form a triangle. Let $BC = a,CA = b,AB = c, PA = x,PB = y, PC = z$, prove that
$$\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c$$
2000 Romania Team Selection Test, 3
Prove that for any positive integers $n$ and $k$ there exist positive integers $a>b>c>d>e>k$ such that
\[n=\binom{a}{3}\pm\binom{b}{3}\pm\binom{c}{3}\pm\binom{d}{3}\pm\binom{e}{3}\]
[i]Radu Ignat[/i]
2017 Canadian Open Math Challenge, C2
Source: 2017 Canadian Open Math Challenge, Problem C2
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A function $f(x)$ is periodic with period $T > 0$ if $f(x + T) = f(x)$ for all $x$. The smallest such number $T$ is called the least period. For example, the functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$.
$\qquad$(a) Let a function $g(x)$ be periodic with the least period $T = \pi$. Determine the least period of $g(x/3)$.
$\qquad$(b) Determine the least period of $H(x) = sin(8x) + cos(4x)$
$\qquad$(c) Determine the least periods of each of $G(x) = sin(cos(x))$ and $F(x) = cos(sin(x))$.
2009 Portugal MO, 1
A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?
1957 Moscow Mathematical Olympiad, 346
Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.
MBMT Team Rounds, 2020.1
Chris has a bag with 4 black socks and 6 red socks (so there are $10$ socks in total). Timothy reaches into the bag and grabs two socks [i]without replacement[/i]. Find the probability that he will not grab two red socks.
[i]Proposed by Chris Tong[/i]
2000 Manhattan Mathematical Olympiad, 4
An equilateral triangle $ABC$ is given, together with a point $P$ inside it.
[asy]
draw((0,0)--(4,0)--(2,3.464)--(0,0));
draw((1.3, 1.2)--(0,0));
draw((1.3, 1.2)--(2,3.464));
draw((1.3, 1.2)--(4,0));
label("$A$",(0,0),SW);
label("$B$",(4,0),SE);
label("$C$",(2,3.464),N);
label("$P$",(1.3,1.2),S);
[/asy]
Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.
LMT Speed Rounds, 2016.3
The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
[i]Proposed by Clive Chan