Found problems: 85335
2020 USMCA, 30
For a positive integer $n$, let $\Omega(n)$ denote the number of prime factors of $n$, counting multiplicity. Let $f_1(n)$ and $f_3(n)$ denote the sum of positive divisors $d|n$ where $\Omega(d)\equiv 1\pmod 4$ and $\Omega(d)\equiv 3\pmod 4$, respectively. For example, $f_1(72) = 72 + 2 + 3 = 77$ and $f_3(72) = 8+12+18 = 38$. Determine $f_3(6^{2020}) - f_1(6^{2020})$.
1975 Canada National Olympiad, 4
For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
1996 Poland - Second Round, 3
$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$
2010 Contests, 4
Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$.
Note: The tiles must completely cover all the board, with no overlappings.
2020 HK IMO Preliminary Selection Contest, 18
Two $n$-sided polygons are said to be of the same type if we can label their vertices in clockwise order as $A_1$, $A_2$, ..., $A_n$ and $B_1$, $B_2$, ..., $B_n$ respectively such that each pair of interior angles $A_i$ and $B_i$ are either both reflex angles or both non-reflex angles. How many different types of $11$-sided polygons are there?
1991 National High School Mathematics League, 2
$a,b,c$ are three non-zero-complex numbers, and $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$, then the value of $\frac{a+b-c}{a-b+c}$ is ($\omega=-\frac{1}{2}+\frac{\sqrt3}{2}\text{i}$)
$\text{(A)}1\qquad\text{(B)}\pm\omega\qquad\text{(C)}1,\omega,\omega^2\qquad\text{(D)}1,-\omega,-\omega^2$
1997 Tournament Of Towns, (552) 2
$M$ is the midpoint of the side $BC$ of a triangle $ABC$. Construct a line $\ell$ intersecting the triangle and parallel to $BC$ such that the segment of $\ell$ between the sides $AB$ and $AC$ is the hypotenuse of a right-angled triangle with $M$ being its third vertex.
(Folklore)
2010 Saudi Arabia Pre-TST, 2.2
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.
PEN N Problems, 8
An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.
2012 Oral Moscow Geometry Olympiad, 1
Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?
2019 Iran MO (2nd Round), 3
$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$
Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$
2018 China Western Mathematical Olympiad, 5
In acute triangle $ABC,$ $AB<AC,$ $O$ is the circumcenter of the triangle. $M$ is the midpoint of segment $BC,$ $(AOM)$ intersects the line $AB$ again at $D$ and intersects the segment $AC$ at $E.$
Prove that $DM=EC.$
2015 BMT Spring, 10
Evaluate
$$\int^{\pi/2}_0\ln(4\sin x)dx.$$
2004 Unirea, 2
Consider a group $ G $ which has the property that any element of it, with the exception of the identity, has order $ p\ge 2. $ Prove that
[b]a)[/b] $ p $ is prime.
[b]b)[/b] $ G $ is commutative if any subset of $ G $ having $ p^2-1 $ elements contains at least $ p $ elements that commute between themselves pairwise.
1991 Putnam, A2
$M$ and $N$ are real unequal $n\times n$ matrices satisfying $M^3=N^3$ and $M^2N=N^2M$. Can we choose $M$ and $N$ so that $M^2+N^2$ is invertible?
2012 Purple Comet Problems, 19
Find the remainder when $2^{5^9}+5^{9^2}+9^{2^5}$ is divided by $11$.
1976 AMC 12/AHSME, 2
For how many real numbers $x$ is $\sqrt{-(x+1)^2}$ a real number?
$\textbf{(A) }\text{none}\qquad\textbf{(B) }\text{one}\qquad\textbf{(C) }\text{two}\qquad\textbf{(D) }\text{a finite number greater than two}\qquad \textbf{(E) }\text{infinitely many}$
2023 USAMTS Problems, 3
We say that three numbers are balanced if either all three numbers are the same, or they are all different. A grid consisting of hexagons is presented in Figure 1. Each hexagon is filled with the number 1, 2, or 3, so that for any three hexagons that are mutually adjacent and oriented with two hexagons on the bottom and one hexagon on the top (as in Figure 3), the three numbers in the hexagons are balanced. Prove that when the grid is filled completely, the three numbers in the three shaded hexagons are balanced.
(An example of a partially filled-in grid is shown in Figure 2. There are other ways of filling in the grid.)
2014 IMO, 1
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
[i]Proposed by Gerhard Wöginger, Austria.[/i]
2008 AMC 12/AHSME, 1
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
$ \textbf{(A)}$ 1:50 PM $ \qquad
\textbf{(B)}$ 3:00 PM $ \qquad
\textbf{(C)}$ 3:30 PM $ \qquad
\textbf{(D)}$ 4:30 PM $ \qquad
\textbf{(E)}$ 5:50 PM
1979 IMO Longlists, 53
An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$,
\[\frac{1}{N}\sum_{n_j\le N} n_j \le c,\]
for every $N >0$.
Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such
that
\[\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}\]
and
\[m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).\]
1982 IMO Longlists, 30
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$
2020 Korean MO winter camp, #6
Find all strictly increasing sequences $\{a_n\}_{n=0}^\infty$ of positive integers such that for all positive integers $k,m,n$
$$\frac{a_{n+1} +a_{n+2} +\dots +a_{n+k}}{k+m}$$ is not an integer larger than $2020$.
2009 National Olympiad First Round, 27
$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$.
$ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$
$\textbf{(A)}\ 1000 \qquad\textbf{(B)}\ 1005 \qquad\textbf{(C)}\ 1010 \qquad\textbf{(D)}\ 2009 \qquad\textbf{(E)}\ 2010$
1978 IMO, 1
In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$