Found problems: 85335
2007 China Team Selection Test, 3
Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.
2017 Bosnia and Herzegovina EGMO TST, 3
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$
2004 Purple Comet Problems, 23
Let $a$ and $b$ be real numbers satisfying \[a^4 + 8b = 4(a^3 - 1) - 16 \sqrt{3}\] and \[b^4 + 8a = 4(b^3 - 1) + 16 \sqrt{3}.\] Find $a^4 + b^4$.
2018 AMC 10, 16
Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$?
$
\textbf{(A) }5 \qquad
\textbf{(B) }8 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }13 \qquad
\textbf{(E) }15 \qquad
$
2020 Bangladesh Mathematical Olympiad National, Problem 6
$f$ is a one-to-one function from the set of positive integers to itself such that $$f(xy) = f(x) × f(y)$$ Find the minimum possible value of $f(2020)$.
2013 BMT Spring, 10
Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms,
$$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$
2014 NIMO Problems, 5
Let $ABC$ be an acute triangle with orthocenter $H$ and let $M$ be the midpoint of $\overline{BC}$. (The [i]orthocenter[/i] is the point at the intersection of the three altitudes.) Denote by $\omega_B$ the circle passing through $B$, $H$, and $M$, and denote by $\omega_C$ the circle passing through $C$, $H$, and $M$. Lines $AB$ and $AC$ meet $\omega_B$ and $\omega_C$ again at $P$ and $Q$, respectively. Rays $PH$ and $QH$ meet $\omega_C$ and $\omega_B$ again at $R$ and $S$, respectively. Show that $\triangle BRS$ and $\triangle CRS$ have the same area.
[i]Proposed by Aaron Lin[/i]
1998 Iran MO (3rd Round), 2
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC.
\]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
\]
2006 Romania National Olympiad, 3
Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.
2013 USAMTS Problems, 5
Let $S$ be a planar region. A $\emph{domino-tiling}$ of $S$ is a partition of $S$ into $1\times2$ rectangles. (For example, a $2\times3$ rectangle has exactly $3$ domino-tilings, as shown below.)
[asy]
import graph; size(7cm);
pen dps = linewidth(0.7); defaultpen(dps);
draw((0,0)--(3,0)--(3,2)--(0,2)--cycle, linewidth(2));
draw((4,0)--(4,2)--(7,2)--(7,0)--cycle, linewidth(2));
draw((8,0)--(8,2)--(11,2)--(11,0)--cycle, linewidth(2));
draw((1,0)--(1,2));
draw((2,1)--(3,1));
draw((0,1)--(2,1), linewidth(2));
draw((2,0)--(2,2), linewidth(2));
draw((4,1)--(7,1));
draw((5,0)--(5,2), linewidth(2));
draw((6,0)--(6,2), linewidth(2));
draw((8,1)--(9,1));
draw((10,0)--(10,2));
draw((9,0)--(9,2), linewidth(2));
draw((9,1)--(11,1), linewidth(2));
[/asy]
The rectangles in the partition of $S$ are called $\emph{dominoes}$.
(a) For any given positive integer $n$, find a region $S_n$ with area at most $2n$ that has exactly $n$ domino-tilings.
(b) Find a region $T$ with area less than $50000$ that has exactly $100002013$ domino-tilings.
1998 North Macedonia National Olympiad, 1
Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$.
Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$
2014-2015 SDML (Middle School), 7
Gizmo is thinking of a geometric sequence in which the third term is $1215$ and the fifth is $540$. Which of the following could be the eighth term of Gizmo's sequence?
$\text{(A) }-160\qquad\text{(B) }-135.5\qquad\text{(C) }216\qquad\text{(D) }240\qquad\text{(E) }472.5$
1988 National High School Mathematics League, 1
Define sequence $(a_n):a_1=1,a_2=2,a_{n+2}=\begin{cases}
5a_{n+1}-3a_n,\text{if }a_n\cdot a_{n+1}\text{ is even}\\
a_{n+1}-a_n,\text{if }a_n\cdot a_{n+1}\text{ is odd}
\end{cases}$
Prove that for all $n\in\mathbb{Z}_+$, $a_n\neq0$.
1976 Chisinau City MO, 124
Find $3$ numbers, each of which is equal to the square of the difference of the other two.
2011 Hanoi Open Mathematics Competitions, 5
Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?
1962 IMO Shortlist, 3
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
1983 AMC 12/AHSME, 29
A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A$, $B$, $C$ and $D$. Also, let the distances from $P$ to $A$, $B$ and $C$, respectively, be $u$, $v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$?
$ \textbf{(A)}\ 1 + \sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{2}\qquad\textbf{(C)}\ 2 + \sqrt{2}\qquad\textbf{(D)}\ 3\sqrt{2}\qquad\textbf{(E)}\ 3 + \sqrt{2}$
2004 Estonia National Olympiad, 3
On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.
Cono Sur Shortlist - geometry, 2009.G4
Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.
1999 All-Russian Olympiad Regional Round, 8.1
A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours
2021-2022 OMMC, 12
Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11$) is $\tfrac ab$ for coprime positive integers $a$, $b$. Find $a + b$.
[i]Proposed by Evan Chang[/i]
2014 China Western Mathematical Olympiad, 7
In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$.
Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]
2004 Romania National Olympiad, 2
Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.
(a) Prove that $f_n$ is well defined.
(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective.
[i]Bogdan Enescu[/i]
Today's calculation of integrals, 869
Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$
Answer the questions below.
(1) Find $I_n.$
(2) Find $\sum_{n=1}^{\infty} I_n.$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$