Found problems: 85335
2023 Austrian MO Regional Competition, 2
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
2017 USAMO, 5
Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which:
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$.
[/list]
[i]Proposed by Ricky Liu[/i]
2008 All-Russian Olympiad, 6
In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.
1970 IMO, 1
$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.
2009 USAMTS Problems, 4
The Rational Unit Jumping Frog starts at $(0, 0)$ on the Cartesian plane, and each minute jumps a distance of exactly $1$ unit to a point with rational coordinates.
(a) Show that it is possible for the frog to reach the point $\left(\frac15,\frac{1}{17}\right)$ in a finite amount of time.
(b) Show that the frog can never reach the point $\left(0,\frac14\right)$.
2024 Euler Olympiad, Round 2, 3
Consider a convex quadrilateral \(ABCD\) with \(AC > BD\). In the plane of this quadrilateral, points \(M\) and \(N\) are chosen such that triangles \(ABM\) and \(CDN\) are equilateral, and segments \(MD\) and \(NA\) intersect lines \(AB\) and \(CD\) respectively. Similarly, points \(P\) and \(Q\) are chosen such that triangles \(ADP\) and \(BCQ\) are equilateral, but here segments \(PB\) and \(QA\) do not intersect lines \(AD\) and \(BC\) respectively.
Prove that \(MN = AC + BD\) if and only if \(PQ = AC - BD\).
[i]Proposed by Zaza Meliqidze, Georgia [/i]
2013 Iran Team Selection Test, 5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
2023 AMC 10, 9
The numbers $16$ and $25$ are a pair of consecutive perfect squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$?
$\textbf{(A) } 674 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1010 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2017$
2012 NIMO Problems, 5
In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3);
draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0));
label("$A$", (.5,.5));
label("$M$", (7/6, 1/6));
label("$N$", (1/3, 4/3));[/asy]
[i]Proposed by Aaron Lin[/i]
MBMT Guts Rounds, 2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes?
[b]D17.[/b] Each digit in a $3$ digit integer is either $1, 2$, or $4$ with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit?
[b]D18 / Z11.[/b] How many two digit numbers are there such that the product of their digits is prime?
[b]D19 / Z9.[/b] In the coordinate plane, a point is selected in the rectangle defined by $-6 \le x \le 4$ and $-2 \le y \le 8$. What is the largest possible distance between the point and the origin, $(0, 0)$?
[b]D20 / Z10.[/b] The sum of two numbers is $6$ and the sum of their squares is $32$. Find the product of the two numbers.
[u]Set 5[/u]
[b]D21 / Z12.[/b] Triangle $ABC$ has area $4$ and $\overline{AB} = 4$. What is the maximum possible value of $\angle ACB$?
[b]D22 / Z13.[/b] Let $ABCD$ be an iscoceles trapezoid with $AB = CD$ and M be the midpoint of $AD$. If $\vartriangle ABM$ and $\vartriangle MCD$ are equilateral, and $BC = 4$, find the area of trapezoid $ABCD$.
[b]D23 / Z14.[/b] Let $x$ and $y$ be positive real numbers that satisfy $(x^2 + y^2)^2 = y^2$. Find the maximum possible value of $x$.
[b]D24 / Z17.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$.
[b]D25.[/b] The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$?
[u]Set 6[/u]
[b]D26 / Z26.[/b] For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get $n$. Estimate the greatest integer $a$ such that $2^a$ evenly divides $n$.
[b]D27 / Z27.[/b] Circles of radius $5$ are centered at each corner of a square with side length $6$. If a random point $P$ is chosen randomly inside the square, what is the probability that $P$ lies within all four circles?
[b]D28 / Z28.[/b] Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s $4$th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class?
[b]D29 / Z29. [/b]Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are $10$ meters west from a roast turkey. Beard, can turn exactly $0.7^o$ and Bored can turn exactly $0.2^o$ degrees. Driving at a consistent $2$ meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey.
Suppose Beard gets to the Turkey in about $818.5$ seconds. Estimate the amount of time it will take Bored.
[b]D30 / Z30.[/b] Let a be the probability that $4$ randomly chosen positive integers have no common divisor except for $1$. Estimate $300a$. Note that the integers $1, 2, 3, 4$ have no common divisor except for $1$.
Remark. This problem is asking you to find $300 \lim_{n\to \infty} a_n$, if $a_n$ is defined to be the probability that $4$ randomly chosen integers from $\{1, 2, ..., n\}$ have greatest common divisor $1$.
PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2011 All-Russian Olympiad Regional Round, 10.5
Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski)
[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url]
Same problem for grades 10 and 11
2001 VJIMC, Problem 4
Let $R$ be an associative non-commutative ring and let $n>2$ be a fixed natural number. Assume that $x^n=x$ for all $x\in R$. Prove that $xy^{n-1}=y^{n-1}x$ holds for all $x,y\in R$.
2016 Saint Petersburg Mathematical Olympiad, 6
The circle contains a closed $100$-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by $720$. Prove that this broken line has an odd number of self-intersection points.
2017 AMC 12/AHSME, 12
What is the sum of the roots of $z^{12} = 64$ that have a positive real part?
$\textbf{(A) }2 \qquad\textbf{(B) }4 \qquad\textbf{(C) }\sqrt{2} +2\sqrt{3}\qquad\textbf{(D) }2\sqrt{2}+ \sqrt{6} \qquad\textbf{(E) }(1 + \sqrt{3}) + (1+\sqrt{3})i$
1973 Swedish Mathematical Competition, 4
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that
\[
\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}
\]
1995 French Mathematical Olympiad, Problem 4
Suppose $A_1,A_2,A_3,B_1,B_2,B_3$ are points in the plane such that for each $i,j\in\{1,2,3\}$ it holds that $A_iB_j=i+j$. What can be said about these six points?
1989 National High School Mathematics League, 10
A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.
2017 CCA Math Bonanza, I1
Find the integer $n$ such that $6!\times7!=n!$.
[i]2017 CCA Math Bonanza Individual Round #1[/i]
Kyiv City MO Juniors Round2 2010+ geometry, 2019.8.41
Through the vertices $A, B$ of the parallelogram $ABCD$ passes a circle that intersects for the second time diagonals $BD$ and $AC$ at points $X$ and $Y$, respectively. The circumsccribed circle of $\vartriangle ADX$ intersects diagonal $AC$ for the second time at the point $Z$. Prove that $AY = CZ$.
2023 Rioplatense Mathematical Olympiad, 5
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all non-negative real number $\alpha$ such that there exist a function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)$$
for any $x,y$ positive real numbers.
2021 BMT, 5
Anthony the ant is at point $A$ of regular tetrahedron $ABCD$ with side length $4$. Anthony wishes to crawl on the surface of the tetrahedron to the midpoint of $\overline{BC}$. However, he does not want to touch the interior of face $\vartriangle ABC$, since it is covered with lava. What is the shortest distance Anthony must travel?
2014 Contests, 1
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
2024 Germany Team Selection Test, 2
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.
Prove that line $AO$ passes through the midpoint of segment $BE$.
2011-2012 SDML (High School), 11
Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$
2001 Regional Competition For Advanced Students, 2
Find all real solutions to the equation
$$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$