Found problems: 85335
2005 AIME Problems, 8
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.
2020 Purple Comet Problems, 28
Let $p, q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.
1996 Vietnam Team Selection Test, 2
There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?
2022 Bulgarian Spring Math Competition, Problem 9.3
Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy
$$\begin{cases}
p + 49 = 2x^2\\
p^2 + 49 = 2y^2\\
\end{cases}$$
STEMS 2024 Math Cat A, P3
Let $ABC$ be a triangle. Let $I$ be the Incenter of $ABC$ and $S$ be the midpoint of arc $BAC$. Define $IA$ as the $A$-excenter wrt $ABC$. Define $\omega$ to be the circle centred at $S$ with radius $SB$. Let $AI_A \cap \omega = X$, $Y$. Show that $\angle BCX = \angle ACY$.
2014 India National Olympiad, 1
In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.
2001 Tournament Of Towns, 7
The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle.
(a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$?
(b) What is the maximum area of this triangle?
2005 Sharygin Geometry Olympiad, 2
Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.
JOM 2015 Shortlist, C8
Let $a$ be a permutation on $\{0,1,\ldots ,2015\}$ and $b,c$ are also permutations on $\{1,2,\ldots ,2015\}$. For all $x\in \{1,2,\ldots ,2015\}$, the following conditions are satisfied:
(i) $a(x)-a(x-1)\neq 1$,\\
(ii) if $b(x)\neq x$, then $c(x)=x$,\\
Prove that the number of $a$'s is equal to the number of ordered pairs of $(b,c)$.
2022-2023 OMMC, 7
Define $\triangle ABC$ with incenter $I$ and $AB=5$, $BC=12$, $CA=13$. A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$'s area divided by $\pi.$
1994 National High School Mathematics League, 1
$a,b,c$ are real numbers. The sufficient and necessary condition of $\forall x\in\mathbb{R},a\sin x+b\cos x+c>0$ is
$\text{(A)}$ $a=b=0,c>0$
$\text{(B)}$ $\sqrt{a^2+b^2}=c$
$\text{(C)}$ $\sqrt{a^2+b^2}<c$
$\text{(D)}$ $\sqrt{a^2+b^2}>c$
2012 Kazakhstan National Olympiad, 2
Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.
2007 AMC 10, 10
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $ 20$, the father is $ 48$ years old, and the average age of the mother and children is $ 16$. How many children are in the family?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
1991 India Regional Mathematical Olympiad, 8
The $64$ squares of an $8 \times 8$ chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares. Show that in fact all the $64$ entries are equal.
2023 International Zhautykov Olympiad, 1
Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$, he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.
2014 Contests, 1
Determine all pairs $(a,b)$ of positive integers satisfying
\[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]
2013 AMC 8, 14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$
1954 Moscow Mathematical Olympiad, 283
Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)
2007 International Zhautykov Olympiad, 2
The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$.
a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$.
b) does it always happen that there exists a right triangle with the above property ?
2017 Brazil National Olympiad, 3.
[b]3.[/b] A quadrilateral $ABCD$ has the incircle $\omega$ and is such that the semi-lines $AB$ and $DC$ intersect at point $P$ and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point of $\omega$ closest from line $PQ$. Prove that the line $RT$ passes through the incenter of triangle $PQC$.
2018 BAMO, 4
(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that
$$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$
(b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)
2005 National Olympiad First Round, 33
Let $K$ be the intersection of diagonals of cyclic quadrilateral $ABCD$, where $|AB|=|BC|$, $|BK|=b$, and $|DK|=d$. What is $|AB|$?
$
\textbf{(A)}\ \sqrt{d^2 + bd}
\qquad\textbf{(B)}\ \sqrt{b^2+bd}
\qquad\textbf{(C)}\ \sqrt{2bd}
\qquad\textbf{(D)}\ \sqrt{2(b^2+d^2-bd)}
\qquad\textbf{(E)}\ \sqrt{bd}
$
1996 AMC 12/AHSME, 20
In the xy-plane, what is the length of the shortest path from $(0, 0)$ to $(12, 16)$ that does not go inside the circle $(x - 6)^2 + (y - 8)^2 = 25$?
$\text{(A)}\ 10\sqrt 3 \qquad \text{(B)}\ 10\sqrt 5 \qquad \text{(C)}\ 10\sqrt 3 + \frac{ 5\pi}{3} \qquad \text{(D)}\ 40\frac{\sqrt{3}}3 \qquad \text{(E)}\ 10+5\pi$
2009 Sharygin Geometry Olympiad, 12
Let $ CL$ be a bisector of triangle $ ABC$. Points $ A_1$ and $ B_1$ are the reflections of $ A$ and $ B$ in $ CL$, points $ A_2$ and $ B_2$ are the reflections of $ A$ and $ B$ in $ L$. Let $ O_1$ and $ O_2$ be the circumcenters of triangles $ AB_1B_2$ and $ BA_1A_2$ respectively. Prove that angles $ O_1CA$ and $ O_2CB$ are equal.
2023 AMC 12/AHSME, 7
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
$\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$