Found problems: 85335
2004 China National Olympiad, 3
Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\cdots +a_{2004}$, where $1\le a_1<a_2<\cdots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$.
[i]Chen Yonggao[/i]
1964 AMC 12/AHSME, 38
The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is:
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$
1979 AMC 12/AHSME, 25
If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by $x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively, when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals
$\textbf{(A) }\frac{1}{256}\qquad\textbf{(B) }-\frac{1}{16}\qquad\textbf{(C) }1\qquad\textbf{(D) }-16\qquad\textbf{(E) }256$
2005 Korea Junior Math Olympiad, 7
If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}}
$$
2001 Moldova National Olympiad, Problem 3
In a triangle $ABC$, the line symmetric to the median through $A$ with respect to the bisector of the angle at $A$ intersects $BC$ at $M$. Points $P$ on $AB$ and $Q$ on $AC$ are chosen such that $MP\parallel AC$ and $MQ\parallel AB$. Prove that the circumcircle of the triangle $MPQ$ is tangent to the line $BC$.
2023 China Second Round, 5
Find the sum of the smallest 20 positive real solutions of the equation $\sin x=\cos 2x .$
2021 Girls in Mathematics Tournament, 1
Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$
1987 Traian Lălescu, 2.2
In a triangle $ ABC $ that has perimeter $ P, $ prove that it's isosceles if and only if
$$ P^2+\sin^2 (\angle ABC-\angle BCA) =4\cdot AB\cdot AC\cdot\cos^2\frac{\angle ABC}{2}\cdot\cos^2\frac{\angle BCA}{2} . $$
2016 Sharygin Geometry Olympiad, 7
Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point.
[i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]
2015 IFYM, Sozopol, 7
A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide.
The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?
ICMC 5, 5
A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down $2/2=1$.
Find the expected sum of all numbers it writes down, given that it is finite.
[i]Proposed by Ethan Tan[/i]
2011 IFYM, Sozopol, 3
If $x$ and $y$ are real numbers, determine the greatest possible value of the expression
$\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.
2020 USMCA, 7
Jenn is competing in a puzzle hunt with six regular puzzles and one additional meta-puzzle. Jenn can solve any puzzle regularly. Additionally, if she has already solved the meta-puzzle, Jenn can also back-solve a puzzle. A back-solve is distinguishable from a regular solve. The meta puzzle cannot be the first puzzle solved. How many possible solve orders for the seven puzzles are possible?
For example, Jenn may solve #3, solve #5, solve #6, solve the meta-puzzle, solve #2, solve #1, and then solve #4.
However, she may not solve #2, solve #4, solve #6, back-solve #1, solve #3, solve #5, and then solve the meta-puzzle.
2022 JHMT HS, 7
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does $2022$ of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
1954 Moscow Mathematical Olympiad, 282
Given a sequence of numbers $a_1, a_2, ..., a_{15}$, one can always construct a new sequence $b_1,b_2, ..., b_{15}$, where $b_i$ is equal to the number of terms in the sequence $\{a_k\}^{15}_{k=1}$ less than $a_i$ ($i = 1, 2,..., 15$). Is there a sequence $\{a_k\}^{15}_{k=1}$ for which the sequence $\{b_k\}^{15}_{k=1}$ is $$1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13 \,?$$
2015 HMIC, 1
Let $S$ be the set of positive integers $n$ such that the inequality
\[
\phi(n) \cdot \tau(n) \geq \sqrt{\frac{n^3}{3}}
\]
holds, where $\phi(n)$ is the number of positive integers $k \le n$ that are relatively prime to $n$, and $\tau(n)$ is the number of positive divisors of $n$. Prove that $S$ is finite.
1950 Putnam, A1
For what values of the ratio $a/b$ is the limaçon $r = a - b \cos \theta$ a convex curve? $(a > b > 0)$
1985 AMC 12/AHSME, 4
A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is
$ \textbf{(A)}\ \$306 \qquad \textbf{(B)}\ \$333 \qquad \textbf{(C)}\ \$342 \qquad \textbf{(D)}\ \$348 \qquad \textbf{(E)}\ \$360$
2009 National Olympiad First Round, 34
$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x \plus{} y^2)(x^2 \minus{} y)/(xy)$ ?
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17$
2024 Romania Team Selection Tests, P1
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.
Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.
[i]Ivan Chan Kai Chin, Malaysia[/i]
2011 All-Russian Olympiad, 4
Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order.
[i]S. Berlov[/i]
2002 China Girls Math Olympiad, 4
Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that \[ \frac{AH}{HF} = \frac{AC}{CD}.\]
2007 Gheorghe Vranceanu, 4
Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as
$$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate:
$$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$
1996 All-Russian Olympiad, 6
In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$.
[i]M. Sonkin[/i]
2019 BMT Spring, Tie 3
We say that a quadrilateral $Q$ is [i]tangential [/i] if a circle can be inscribed into it, i.e. there exists a circle $C$ that does not meet the vertices of $Q$, such that it meets each edge at exactly one point. Let $N$ be the number of ways to choose four distinct integers out of $\{1, . . . , 24\}$ so that they form the side lengths of a tangential quadrilateral. Find the largest prime factor of $N$.