Found problems: 85335
Oliforum Contest I 2008, 3
Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$.
Prove that $ AA',BB'$ and $ CC'$ are concurrent.
[img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]
1997 May Olympiad, 3
There are $10000$ equal tiles in the shape of an equilateral triangle. With these little triangles, regular hexagons are formed, without overlaps or gaps. If the regular hexagon that wastes the fewest triangles is formed, how many triangles are left over?
1916 Eotvos Mathematical Competition, 3
Divide the numbers
$$1, 2,3, 4,5$$
into two arbitrarily chosen sets. Prove that one of the sets contains two numbers and their difference.
2017 ASDAN Math Tournament, 7
Point $C$ is chosen on the arc of a semicircle with diameter $AB$. The two circles with diameters of $AC$ and $BC$ intersect again at point $D$. If $DA=20$ and $DB=16$, compute the length of $DC$.
2020 BMT Fall, 19
Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.)
[img]https://cdn.artofproblemsolving.com/attachments/5/1/26e8aa6d12d9dd85bd5b284b6176870c7d11b1.png[/img]
2022 Mexican Girls' Contest, 2
Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying
$$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$
Find all possible values of $\angle BCP$.
1977 All Soviet Union Mathematical Olympiad, 244
Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also.
a) Find all two- and four-digit fine numbers.
b) Is there any six-digit fine number?
c) Prove that there exists $20$-digit fine number.
d) Prove that there exist at least ten $100$-digit fine numbers.
e) Prove that there exists $30$-digit fine number.
2012 Math Prize for Girls Olympiad, 1
Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that
\[
\sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, .
\]
2006 Princeton University Math Competition, 10
What is the largest possible number of vertices one can have in a graph that satisfies the following conditions: each vertex is connected to exactly $3$ other vertices, and there always exists a path of length less than or equal to $2$ between any two vertices?
2007 ITAMO, 2
We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q.
a) if P,Q are similar, then $P(2007)-Q(2007)$ is even
b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?
2021 Indonesia MO, 5
Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.
2025 Chile TST IMO-Cono, 4
Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).
1995 VJIMC, Problem 3
Let $f(x)$ and $g(x)$ be mutually inverse decreasing functions on the interval $(0,\infty)$. Can it hold that $f(x)>g(x)$ for all $x\in(0,\infty)$?
2022 Romania National Olympiad, P4
Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$
[i]Marius Mînea[/i]
2019 AMC 12/AHSME, 1
Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?
$\textbf{(A) } \frac{5}{8} \qquad \textbf{(B) } \frac{4}{5} \qquad \textbf{(C) } \frac{7}{8} \qquad \textbf{(D) } \frac{9}{10} \qquad \textbf{(E) } \frac{11}{12}$
PEN H Problems, 13
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]
2023 Saint Petersburg Mathematical Olympiad, 7
Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, what is the minimal number of connected components of the graph $G/(X \cup Y)$?
2013 Princeton University Math Competition, 14
Shuffle a deck of $71$ playing cards which contains $6$ aces. Then turn up cards from the top until you see an ace. What is the average number of cards required to be turned up to find the first ace?
2017 Yasinsky Geometry Olympiad, 5
The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.
2001 Estonia National Olympiad, 2
Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.
2020 Korean MO winter camp, #1
Call a positive integer [i]challenging[/i] if it can be expressed as $2^a(2^b+1)$, where $a,b$ are positive integers. Prove that if $X$ is a set of challenging numbers smaller than $2^n (n$ is a given positive integer) and $|X|\ge \frac{4}{3}(n-1)$, there exist two disjoint subsets $A,B\subset X$ such that $|A|=|B|$ and $\sum_{a\in A}a=\sum_{b \in B}b$.
2009 Tournament Of Towns, 5
In rhombus $ABCD$, angle $A$ equals $120^o$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ so that angle $NAM$ equals $30^o$. Prove that the circumcenter of triangle $NAM$ lies on a diagonal of of the rhombus.
2020 Bosnia and Herzegovina Junior BMO TST, 1
Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds:
$\overline{ab}=3 \cdot \overline{cd} + 1$.
2010 South East Mathematical Olympiad, 3
The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$
2015 China Team Selection Test, 3
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.