Found problems: 85335
2013 Indonesia MO, 6
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.
2021-IMOC, G1
Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.
1996 IMO Shortlist, 9
Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$
\[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\]
1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained.
2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?
2005 Today's Calculation Of Integral, 50
Let $a,b$ be real numbers such that $a<b$.
Evaluate
\[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].
2005 Taiwan TST Round 1, 3
Find all positive integer triples $(x,y,z)$ such that
$x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$.
Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)
2012 JHMT, 4
Circle $O$ has radius $18$. From diameter $AB$, there exists a point $C$ such that $BC$ is tangent to $O$ and $AC$ intersects $O$ at a point $D$, with $AD = 24$. What is the length of $BC$?
1982 Czech and Slovak Olympiad III A, 2
Given real numbers $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$. Let $M$ denote the maximum of their absolute values. Prove that it is valid $$
| x_1x_4-x_1x_5 +x_2x_5 -x_2x_6+x_3x_6-x_3x_4| \le 4M^2$$
1993 AIME Problems, 12
The vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?
1986 AMC 8, 6
$ \frac{2}{1\minus{}\frac{2}{3}}\equal{}$
\[ \textbf{(A)}\ \minus{}3 \qquad
\textbf{(B)}\ \minus{}\frac{4}{3} \qquad
\textbf{(C)}\ \frac{2}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 6
\]
2018 Sharygin Geometry Olympiad, 1
Let $M$ be the midpoint of $AB$ in a right angled triangle $ABC$ with $\angle C = 90^\circ$. A circle passing through $C$ and $M$ meets segments $BC, AC$ at $P, Q$ respectively. Let $c_1, c_2$ be the circles with centers $P, Q$ and radii $BP, AQ$ respectively. Prove that $c_1, c_2$ and the circumcircle of $ABC$ are concurrent.
2015 Indonesia MO, 3
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.
1988 IMO Longlists, 89
We match sets $ M$ of points in the coordinate plane to sets $ M*$ according to the rule that $ (x*,y*) \in M*$ if and only if $ x \cdot x* \plus{} y \cdot y* \leq 1$ whenever $ (x,y) \in M.$ Find all triangles $ Q$ such that $ Q*$ is the reflection of $ Q$ in the origin.
1978 Chisinau City MO, 166
It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.
2021 China Girls Math Olympiad, 2
In acute triangle $ABC$ ($AB \neq AC$), $I$ is its incenter and $J$ is the $A$-excenter. $X, Y$ are on minor arcs $\widehat{AB}$ and $\widehat{AC}$ respectively such that $\angle{AXI}=\angle{AYJ}=90^{\circ}$. $K$ is on line $BC$ such that $KI=KJ$.
Proof that line $AK$ bisects $\overline{XY}$.
2016 Peru Cono Sur TST, P5
Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square.
2023 USA TSTST, 6
Let $ABC$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle PAQ,\,\angle PBQ,$ and $\angle PCQ$ are the altitudes of triangle $ABC$. Prove that the midpoint of $\overline{PQ}$ lies on the Euler line of $ABC$.
(The Euler line is the line through the circumcenter and orthocenter of a triangle.)
[i]Proposed by Holden Mui[/i]
2023 AIME, 8
Rhombus $ABCD$ has $\angle BAD<90^{\circ}$. There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to lines $DA$, $AB$, and $BC$ are $9$, $5$, and $16$, respectively. Find the perimeter of $ABCD$.
2011 District Round (Round II), 4
Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A [i]move[/i] consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
2019 Math Prize for Girls Problems, 5
Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$ and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares. What is the total area of these squares?
2001 National Olympiad First Round, 20
If the sum of any $10$ of $21$ real numbers is less than the sum of remaining $11$ of them, at least how many of these $21$ numbers are positive?
$
\textbf{(A)}\ 18
\qquad\textbf{(B)}\ 19
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2014 China Team Selection Test, 3
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$
2011 Greece JBMO TST, 1
a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$.
b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$
2007 Junior Balkan Team Selection Tests - Moldova, 6
The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$
KoMaL A Problems 2019/2020, A. 768
Let $S$ be a shape in the plane which is obtained as a union of finitely many unit squares. Prove that the ratio of the perimeter and the area of $S$ is at most $8$.
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]