Found problems: 85335
2020-2021 Fall SDPC, 7
Alice is wandering in the country of Wanderland. Wanderland consists of a finite number of cities, some of which are connected by two-way trains, such that Wanderland is connected: given any two cities, there is always a way to get from one to the other through a series of train rides.
Alice starts at Riverbank City and wants to end up at Conscious City. Every day, she picks a train going out of the city she is in uniformly at random among all of the trains, and then boards that train to the city it leads to. Show that the expected number of days it takes for her to reach Conscious City is finite.
2018 Belarusian National Olympiad, 10.5
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.
2002 Putnam, 3
Show that for all integers $n>1$, \[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]
2014 AMC 10, 8
Which of the following numbers is a perfect square?
$ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $
1992 AMC 8, 5
A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy]
fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray);
draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1));
fill(circle((1,5/4),1/2),white);
draw(circle((1,5/4),1/2),linewidth(1));
[/asy]
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$
1951 Miklós Schweitzer, 9
Let $ \{m_1,m_2,\dots\}$ be a (finite or infinite) set of positive integers. Consider the system of congruences
(1) $ x\equiv 2m_i^2 \pmod{2m_i\minus{}1}$ ($ i\equal{}1,2,...$ ).
Give a necessary and sufficient condition for the system (1) to be solvable.
2022 Utah Mathematical Olympiad, 4
Alpha and Beta are playing a game on a $10\times 100$ grid of squares. At each turn, they can fold the grid along any of the interior horizontal or vertical gridlines, which creates a smaller (folded) grid of squares (on the first move, they can choose one of $9$ horizontal or $99$ vertical gridlines). The person who makes the last fold wins. If both players play optimally and Alpha starts, determine with proof who wins.
2014 Online Math Open Problems, 19
In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$.
[i]Proposed by Ray Li[/i]
2017 Harvard-MIT Mathematics Tournament, 26
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at $0$. Every minute, he has a $\frac{1}{3}$ chance of moving $1$ unit left, a $\frac{1}{3}$ chance of moving $1$ unit right, and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to $0$ (not including the start) before he gets eaten.
1996 Miklós Schweitzer, 5
Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$.
2010 Saudi Arabia BMO TST, 4
Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.
1996 Abels Math Contest (Norwegian MO), 4
Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$.
Find all possible values of $f(1995)$.
2004 Croatia National Olympiad, Problem 1
Parts of a pentagon have areas $x,y,z$ as shown in the picture. Given the area $x$, find the areas $y$ and $z$ and the area of the entire pentagon.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS9mLzM5NjNjNDcwY2ZmMzgzY2QwYWM0YzI1NmYzOWU2MWY1NTczZmYxLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wOCBhdCA0LjMwLjU1IFBNLnBuZw[/img]
2002 India Regional Mathematical Olympiad, 4
Suppose the integers $1,2,\ldots 10$ are split into two disjoint collections $a_1,a_2, \ldots a_5$ and $b_1 , \ldots b_5$ such that $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$
(i) Show that the larger number in any pair $\{ a_j, b_j \}$ , $1 \leq j \leq 5$ is at least $6$.
(ii) Show that $\sum_{i=1} ^{5} | a_i - b_i|$ = 25 for every such partition.
2010 Iran MO (3rd Round), 4
in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)
2015 India Regional MathematicaI Olympiad, 6
Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer.
(Here $\{a\}$ denotes the fractional part of $a$.)
1935 Moscow Mathematical Olympiad, 003
The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.
2002 All-Russian Olympiad Regional Round, 11.6
There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?
2003 Manhattan Mathematical Olympiad, 2
A tennis net is made of strings tied up together which make a grid consisting of small congruent squares as shown below.
[asy]
size(500);
xaxis(-50,50);
yaxis(-5,5);
add(shift(-50,-5)*grid(100,10));[/asy]
The size of the net is $100\times 10$ small squares. What is the maximal number of edges of small squares which can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends.)
2016 CCA Math Bonanza, L1.3
If the GCD of $a$ and $b$ is $12$ and the LCM of $a$ and $b$ is $168$, what is the value of $a\times b$?
[i]2016 CCA Math Bonanza L1.3[/i]
2008 ITest, 94
Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[\left\lfloor\dfrac{2008}1\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^2}2\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^3}3\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^4}4\right\rfloor,\,\,\,\,\,\,\,\,\,\ldots.\]
2014 Belarusian National Olympiad, 8
An $n\times n$ square is divided into $n^2$ unit cells. Is it possible to cover this square with some layers of 4-cell figures of the following shape [img]https://cdn.artofproblemsolving.com/attachments/5/7/d42a8011ec4c5c91c337296d8033d412fade5c.png[/img](i.e. each cell of the square must be covered with the same number of these figures) if
a) $n=6$?
b) $n=7$?
(The sides of each figure must coincide with the sides of the cells; the figures may be rotated and turned over, but none of them can go beyond the bounds of the square.)
2020 MOAA, TO4
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
Ukrainian TYM Qualifying - geometry, 2013.9
Given a triangle $PQR$, the inscribed circle $\omega$ which touches the sides $QR, RP$ and $PQ$ at points $A, B$ and $C$, respectively, and $AB^2 + AC^2 = 2BC^2$. Prove that the point of intersection of the segments $PA, QB$ and $RC$, the center of the circle $\omega$, the point of intersection of the medians of the triangle $ABC$, the point $A$ and the midpoints of the segments $AC$ and $AB$ lie on one circle.
2024 CIIM, 4
Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit:
\[
\lim_{n \to \infty} \frac{S(n)}{n^3}.
\]