Found problems: 85335
2020 LMT Fall, A26
Jeff has planted $7$ radishes, labelled $R$, $A$, $D$, $I$, $S$, $H$, and $E$. Taiki then draws circles through $S,H,I,E,D$, then through $E,A,R,S$, and then through $H,A,R,D$, and notices that lines drawn through $SH$, $AR$, and $ED$ are parallel, with $SH = ED$. Additionally, $HER$ is equilateral, and $I$ is the midpoint of $AR$. Given that $HD = 2$, $HE$ can be written as $\frac{-\sqrt{a} + \sqrt{b} + \sqrt{1+\sqrt{c}}}{2}$, where $a,b,$ and $c$ are integers, find $a+b+c$.
[i]Proposed by Jeff Lin[/i]
1993 Iran MO (2nd round), 3
Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that
\[ 1 < \frac ab < 1 +\frac 1n.\]
Durer Math Competition CD Finals - geometry, 2008.D1
Given a square grid where the distance between two adjacent grid points is $1$. Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?
1956 AMC 12/AHSME, 9
Simplify $ \left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$; the result is:
$ \textbf{(A)}\ a^{16} \qquad\textbf{(B)}\ a^{12} \qquad\textbf{(C)}\ a^8 \qquad\textbf{(D)}\ a^4 \qquad\textbf{(E)}\ a^2$
2007 Turkey Team Selection Test, 3
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]
2002 Romania National Olympiad, 1
Let $A$ be a ring.
$a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$.
$b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.
2007 Tuymaada Olympiad, 3
$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are altitudes of an acute triangle $ ABC$. A circle passing through $ A_{1}$ and $ B_{1}$ touches the arc $ AB$ of its circumcircle at $ C_{2}$. The points $ A_{2}$, $ B_{2}$ are defined similarly. Prove that the lines $ AA_{2}$, $ BB_{2}$, $ CC_{2}$ are concurrent.
2012 Greece JBMO TST, 1
Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$
2020 ELMO Problems, P3
Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.
[i]Proposed by Fedir Yudin.[/i]
2000 Harvard-MIT Mathematics Tournament, 2
Simplify $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^6+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^6$ to the form $a+bi$.
2023 India IMO Training Camp, 2
Let $\mathbb R^+$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ satisfying \[f(x+y^2f(x^2))=f(xy)^2+f(x)\] for all $x,y \in \mathbb{R}^+$.
[i]Proposed by Shantanu Nene[/i]
1997 IMO Shortlist, 6
(a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that
\[ x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}.\]
(b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that
\[ x^a \plus{} y^b \equal{} z^c.\]
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$
2007 AMC 10, 7
Last year Mr. John Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he has left in state taxes. He paid a total of $ \$10,500$ for both taxes. How many dollars was the inheritance?
$ \textbf{(A)}\ 30,000 \qquad \textbf{(B)}\ 32,500 \qquad \textbf{(C)}\ 35,000 \qquad \textbf{(D)}\ 37,500 \qquad \textbf{(E)}\ 40,000$
2015 IFYM, Sozopol, 1
Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.
2015 QEDMO 14th, 1
Let $n$ be a natural number. A regular hexagon with edge length $n$ gets split into equilateral exploded triangles whose edges are $1$ in length and parallel to one side of the hexagon. Find the number of regular hexagons, the angles of which are all angles of these triangles are.
2024 UMD Math Competition Part II, #5
Define two sequences $x_n, y_n$ for $n = 1, 2, \ldots$ by \[x_n = \left(\sum^n_{k=0} \binom{2n}{2k}49^k 48^{n-k} \right) -1, \quad \text{and} \quad y_n = \sum^{n-1}_{k=0} \binom{2n}{2k + 1} 49^k 48^{n-k}\] Prove there is a positive integer $m$ for which for every integer $n > m,$ the greatest common factor of $x_n$ and $y_n$ is more than $10^{2024}.$
1986 Tournament Of Towns, (126) 1
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
2012 International Zhautykov Olympiad, 3
Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.
2022 Putnam, A3
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3,\ldots$ such that $a_n \in \{1, 2,\ldots, p-1\}$ and $a_na_{n+2}\equiv1+a_{n+1}$ (mod $p$) for all $n\geq 1.$ Prove that $f(p)$ is congruent to 0 or 2 (mod 5).
2016 ASDAN Math Tournament, 10
A point $P$ and a segment $AB$ with length $20$ are randomly drawn on a plane. Suppose that the probability that a randomly selected line passing through $P$ intersects segment $AB$ is $\tfrac{1}{2}$. Next, randomly choose point $Q$ on segment $AB$. What is the probability with respect to choosing $Q$ that a circle centered at $Q$ passing through $P$ contains both $A$ and $B$ in its interior?
2010 Singapore MO Open, 2
Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$
$a_{n+1} = 7a_n + 12b_n + 6$
$b_{n+1} = 4a_n + 7b_n + 3$
Prove that $a_n^2$ is the difference of two consecutive cubes.
2003 AMC 8, 21
The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters?
[asy]/* AMC8 2003 #21 Problem */
size(4inch,2inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);[/asy]
$ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$
2021 LMT Spring, A30
Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Aditya Rao[/i]
2006 Petru Moroșan-Trident, 1
What relationship should be between the positive real numbers $ a $ and $ b $ such that the sequence $ \left(\left( a\sqrt[n]{n} +b \right)^{\frac{n}{\ln n}}\right)_{n\ge 1} $ has a nonzero and finite limit? For such $ a,b, $ calculate the limit of this sequence.
[i]Ion Cucurezeanu[/i]