Found problems: 85335
2023 NMTC Junior, P4
There are $n$ (an even number) bags. Each bag contains atleast one apple and at most $n$ apples. The total number of apples is $2n$. Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is $n$.
1950 AMC 12/AHSME, 16
The number of terms in the expansion of $ [(a\plus{}3b)^2(a\minus{}3b)^2]^2$ when simplified is:
$\textbf{(A)}\ 4\qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
2004 Swedish Mathematical Competition, 3
A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .
1994 AMC 8, 14
Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is
$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 36$
2017 CCA Math Bonanza, I8
Let $a_1,a_2,\ldots,a_{18}$ be a list of prime numbers such that $\frac{1}{64}\times a_1\times a_2\times\cdots\times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18}\frac{1}{\log_{a_i}n}$$ is a positive integer.
[i]2017 CCA Math Bonanza Individual Round #8[/i]
1961 AMC 12/AHSME, 1
When simplified, $(-\frac{1}{125})^{-2/3}$ becomes:
${{ \textbf{(A)}\ \frac{1}{25} \qquad\textbf{(B)}\ -\frac{1}{25} \qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ -25}\qquad\textbf{(E)}\ 25\sqrt{-1}} $
2002 India IMO Training Camp, 7
Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.
2022 Indonesia MO, 5
Let $N\ge2$ be a positive integer. Given a sequence of natural numbers $a_1,a_2,a_3,\dots,a_{N+1}$ such that for every integer $1\le i\le j\le N+1$,
$$a_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N}$$
Prove that there exist a positive integer $k\le N+1$ such that $\gcd(a_k, N) \neq 1$
1974 IMO Longlists, 42
In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.
2016 Online Math Open Problems, 6
In a round-robin basketball tournament, each basketball team plays every other basketball team exactly once. If there are $20$ basketball teams, what is the greatest number of basketball teams that could have at least $16$ wins after the tournament is completed?
[i]Proposed by James Lin[/i]
2021 Kyiv Mathematical Festival, 4
Find all collections of $63$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal. (O. Rudenko)
2023 Bulgaria National Olympiad, 2
Let $ABC$ be an acute triangle and $A_{1}, B_{1}, C_{1}$ be the touchpoints of the excircles with the segments $BC, CA, AB$ respectively. Let $O_{A}, O_{B}, O_{C}$ be the circumcenters of $\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1}$ respectively. Prove that the lines through $O_{A}, O_{B}, O_{C}$ respectively parallel to the internal angle bisectors of $\angle A,\angle B, \angle C$ are concurrent.
2013 Princeton University Math Competition, 3
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways can he get to McCosh without going through Frist?
2014 Postal Coaching, 1
Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.
2007 Miklós Schweitzer, 3
Denote by $\omega (n)$ the number of prime divisors of the natural number $n$ (without multiplicities). Let
$$F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)$$
Prove that $G(x)-F(x)\to \infty$ as $x\to\infty$.
(translated by Miklós Maróti)
2016 AMC 12/AHSME, 10
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$
1997 Chile National Olympiad, 7
In a career in mathematics, $7$ courses are taught, among which students can choose the ones you want. Determine the number of students in the career, knowing that:
$\bullet$ No two students have chosen the same courses.
$\bullet$ Any two students have at least one course in common.
$\bullet$ If the race had one more student, it would not be possible to do both.
2016 Argentina National Olympiad, 5
Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.
1983 Czech and Slovak Olympiad III A, 6
Consider a circle $k$ with center $S$ and radius $r$. Denote $\mathsf M$ the set of all triangles with incircle $k$ such that the largest inner angle is twice bigger than the smallest one. For a triangle $\mathcal T\in\mathsf M$ denote its vertices $A,B,C$ in way that $SA\ge SB\ge SC$. Find the locus of points $\{B\mid\mathcal T\in\mathsf M\}$.
2022 Iran Team Selection Test, 5
Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic.
Proposed by Navid Safaei
2006 Putnam, B6
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for $n\ge 0.$ Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]
2010 Macedonia National Olympiad, 4
The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.
2019 MIG, 13
What is the remainder when $1 + 10 + 19 + 28 + \cdots + 91$ is divided by $9$?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }8$
2014 IMO Shortlist, C9
There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$.
Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd.
[i]Proposed by Tejaswi Navilarekallu, India[/i]
1939 Moscow Mathematical Olympiad, 044
Prove that $cos \frac{2\pi}{5} +cos \frac{4\pi}{5} = -\frac{1}{2}$.