Found problems: 85335
2021 Germany Team Selection Test, 3
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
$\emph{Slovakia}$
2006 ITAMO, 5
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
a) Find all $n\ge 3$ such that the inequality is true for positive reals.
b) Find all $n\ge 3$ such that the inequality is true for reals.
2006 Federal Math Competition of S&M, Problem 2
For an arbitrary point $M$ inside a given square $ABCD$, let $T_1,T_2,T_3$ be the centroids of triangles $ABM,BCM$, and $DAM$, respectively. Let $OM$ be the circumcenter of triangle $T_1T_2T_3$. Find the locus of points $OM$ when $M$ takes all positions within the interior of the square.
2023 Euler Olympiad, Round 1, 9
Let's call the positive integer $x$ interesting, if there exists integer $y$ such that the following equation holds: $(x + y)^y = (x - y)^x.$ Suppose we list all interesting integers in increasing order. An interesting integer is considered very interesting if it is not relatively prime with any other interesting integer preceding it. Find the second very interesting integer.
[i]Note: It is assumed that the first interesting integer is not very interesting.[/i]
[i]Proposed by Zurab Aghdgomelashvili, Georgia[/i]
MMATHS Mathathon Rounds, 2018
[u]Round 5 [/u]
[b]p13.[/b] Circles $\omega_1$, $\omega_2$, and $\omega_3$ have radii $8$, $5$, and $5$, respectively, and each is externally tangent to the other two. Circle $\omega_4$ is internally tangent to $\omega_1$, $\omega_2$, and $\omega_3$, and circle $\omega_5$ is externally tangent to the same three circles. Find the product of the radii of $\omega_4$ and $\omega_5$.
[b]p14.[/b] Pythagoras has a regular pentagon with area $1$. He connects each pair of non-adjacent vertices with a line segment, which divides the pentagon into ten triangular regions and one pentagonal region. He colors in all of the obtuse triangles. He then repeats this process using the smaller pentagon. If he continues this process an infinite number of times, what is the total area that he colors in? Please rationalize the denominator of your answer.
p15. Maisy arranges $61$ ordinary yellow tennis balls and $3$ special purple tennis balls into a $4 \times 4 \times 4$ cube. (All tennis balls are the same size.) If she chooses the tennis balls’ positions in the cube randomly, what is the probability that no two purple tennis balls are touching?
[u]Round 6 [/u]
[b]p16.[/b] Points $A, B, C$, and $D$ lie on a line (in that order), and $\vartriangle BCE$ is isosceles with $\overline{BE} = \overline{CE}$. Furthermore, $F$ lies on $\overline{BE}$ and $G$ lies on $\overline{CE}$ such that $\vartriangle BFD$ and $\vartriangle CGA$ are both congruent to $\vartriangle BCE$. Let $H$ be the intersection of $\overline{DF}$ and $\overline{AG}$, and let $I$ be the intersection of $\overline{BE}$ and $\overline{AG}$. If $m \angle BCE = arcsin \left( \frac{12}{13} \right)$, what is $\frac{\overline{HI}}{\overline{FI}}$ ?
[b]p17.[/b] Three states are said to form a tri-state area if each state borders the other two. What is the maximum possible number of tri-state areas in a country with fifty states? Note that states must be contiguous and that states touching only at “corners” do not count as bordering.
[b]p18.[/b] Let $a, b, c, d$, and $e$ be integers satisfying $$2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c +\sqrt[3]{2}d + e = 0$$ and $$25\sqrt5 i + 25a - 5\sqrt5 ib - 5c + \sqrt5 id + e = 0$$ where $i =\sqrt{-1}$. Find $|a + b + c + d + e|$.
[u]Round 7[/u]
[b]p19.[/b] What is the greatest number of regions that $100$ ellipses can divide the plane into? Include the unbounded region.
[b]p20.[/b] All of the faces of the convex polyhedron $P$ are congruent isosceles (but NOT equilateral) triangles that meet in such a way that each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. (An isosceles triangle has two base angles and one vertex angle.) Find the sum of the numbers of faces, edges, and vertices of $P$.
[b]p21.[/b] Find the number of ordered $2018$-tuples of integers $(x_1, x_2, .... x_{2018})$, where each integer is between $-2018^2$ and $2018^2$ (inclusive), satisfying $$6(1x_1 + 2x_2 +...· + 2018x_{2018})^2 \ge (2018)(2019)(4037)(x^2_1 + x^2_2 + ... + x^2_{2018}).$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2784936p24472982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1966 AMC 12/AHSME, 11
The sides of triangle $BAC$ are in the ratio $2: 3: 4$. $BD$ is the angle-bisector drawn to the shortest side $AC$, dividing it into segments $AD$ and $CD$. If the length of $AC$ is $10$, then the length of the longer segment of $AC$ is:
$\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12$
1996 IMC, 3
The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if
$A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$.
i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors
of $A$.
ii) Find the maximal number of distinct pairwise commuting involutions on $V$.
2002 AMC 12/AHSME, 24
A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$.
$ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$
$ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$
2008 Baltic Way, 17
Assume that $ a$, $ b$, $ c$ and $ d$ are the sides of a quadrilateral inscribed in a given circle. Prove that the product $ (ab \plus{} cd)(ac \plus{} bd)(ad \plus{} bc)$ acquires its maximum when the quadrilateral is a square.
2022 Sharygin Geometry Olympiad, 7
A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$.
Restore the triangle.
2006 AMC 12/AHSME, 16
Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area?
$ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$
2014 BMT Spring, 13
Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.
2024 USAMTS Problems, 5
Find all ordered triples of nonnegative integers $(a,b,c)$ satisfying $2^a \cdot 5^b - 3^c = 1.$
2013 Online Math Open Problems, 6
Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
[i]Ray Li[/i]
1992 French Mathematical Olympiad, Problem 1
Let $\Delta$ be a convex figure in a plane $\mathcal P$. Given a point $A\in\mathcal P$, to each pair $(M,N)$ of points in $\Delta$ we associate the point $m\in\mathcal P$ such that $\overrightarrow{Am}=\frac{\overrightarrow{MN}}2$ and denote by $\delta_A(\Delta)$ the set of all so obtained points $m$.
(a)
i. Prove that $\delta_A(\Delta)$ is centrally symmetric.
ii. Under which conditions is $\delta_A(\Delta)=\Delta$?
iii. Let $B,C$ be points in $\mathcal P$. Find a transformation which sends $\delta_B(\Delta)$ to $\delta_C(\Delta)$.
(b) Determine $\delta_A(\Delta)$ if
i. $\Delta$ is a set in the plane determined by two parallel lines.
ii. $\Delta$ is bounded by a triangle.
iii. $\Delta$ is a semi-disk.
(c) Prove that in the cases $b.2$ and $b.3$ the lengths of the boundaries of $\Delta$ and $\delta_A(\Delta)$ are equal.
2001 India Regional Mathematical Olympiad, 4
Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.
1957 AMC 12/AHSME, 2
In the equation $ 2x^2 \minus{} hx \plus{} 2k \equal{} 0$, the sum of the roots is $ 4$ and the product of the roots is $ \minus{}3$. Then $ h$ and $ k$ have the values, respectively:
$ \textbf{(A)}\ 8\text{ and }{\minus{}6} \qquad
\textbf{(B)}\ 4\text{ and }{\minus{}3}\qquad
\textbf{(C)}\ {\minus{}3}\text{ and }4\qquad
\textbf{(D)}\ {\minus{}3}\text{ and }8\qquad
\textbf{(E)}\ 8\text{ and }{\minus{}3}$
1955 AMC 12/AHSME, 27
If $ r$ and $ s$ are the roots of $ x^2\minus{}px\plus{}q\equal{}0$, then $ r^2\plus{}s^2$ equals:
$ \textbf{(A)}\ p^2\plus{}2q \qquad
\textbf{(B)}\ p^2\minus{}2q \qquad
\textbf{(C)}\ p^2\plus{}q^2 \qquad
\textbf{(D)}\ p^2\minus{}q^2 \qquad
\textbf{(E)}\ p^2$
2020 EGMO, 4
A permutation of the integers $1, 2, \ldots, m$ is called [i]fresh[/i] if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$.
Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$.
[i]For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.[/i]
2016 China Team Selection Test, 5
Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form
$$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$
,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?
1985 Iran MO (2nd round), 4
Let $x$ and $y$ be two real numbers. Prove that the equations
\[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\]
Holds if and only if at least one of $x$ or $y$ be integer.
2011 Philippine MO, 2
In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.
2005 MOP Homework, 4
Let $x_1$, $x_2$, ..., $x_5$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5=5$. Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.
2017 China National Olympiad, 6
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$
2009 Kosovo National Mathematical Olympiad, 2
Solve the equation:
$x^2+2xcos(x-y)+1=0$