Found problems: 85335
2018 China Northern MO, 5
A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.
2021 AMC 10 Fall, 22
Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres each with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r?$
$\textbf{(A) }\dfrac32\qquad\textbf{(B) }\dfrac{90 - 40\sqrt3}{11}\qquad\textbf{(C) }2\qquad\textbf{(D) }\dfrac{144 - 25\sqrt3}{44}\qquad\textbf{(E) }\dfrac52$
2013 USAMO, 4
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
2021 Yasinsky Geometry Olympiad, 1
A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm . Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$.
(Alexey Panasenko)
2021 China Second Round Olympiad, Problem 12
Let $C$ be the left vertex of the ellipse $\frac{x^2}8+\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$.
(i) Compute the maximum of $|CA|+|CB|$.
(ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and $N$, respectively. If the intersection of the perpendicular bisector of $MN$ and the circle with diameter $MN$ lies inside the given ellipse, compute the range of possible values of $k$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 12)[/i]
2023 MOAA, 1
Find the last digit of $2023^{2023}$.
[i]Proposed by Yifan Kang[/i]
1989 China National Olympiad, 4
Given a triangle $ABC$, points $D,E,F$ lie on sides $BC,CA,AB$ respectively. Moreover, the radii of incircles of $\triangle AEF, \triangle BFD, \triangle CDE$ are equal to $r$. Denote by $r_0$ and $R$ the radii of incircles of $\triangle DEF$ and $\triangle ABC$ respectively. Prove that $r+r_0=R$.
2012 All-Russian Olympiad, 3
Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.
2022 AMC 12/AHSME, 13
The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
[asy]
size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4));
draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4));
[/asy]
$\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$
1967 Putnam, A6
Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations
$$ a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0 $$
for which no $x_i$ is zero. Each such solution generates a $4$-tuple of plus and minus signs (by considering the sign of $x_i$).
[list=a]
[*] Determine, with proof, the maximum number of distinct $4$-tuples possible.
[*] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$-tuples is attained.
2014 HMNT, 9
How many lines pass through exactly two points in the following hexagonal grid?
[img]https://cdn.artofproblemsolving.com/attachments/2/e/35741c80d0e0ee0ca56f1297b1e377c8db9e22.png[/img]
2007 Stanford Mathematics Tournament, 5
The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?
2015 AMC 12/AHSME, 16
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron?
$\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$
2025 Malaysian IMO Training Camp, 1
Given two primes $p$ and $q$, is $v_p(q^n+n^q)$ unbounded as $n$ varies?
[i](Proposed by Ivan Chan Kai Chin)[/i]
2018 Pan-African Shortlist, C5
A set of $n$ lines are said to be in [i]standard form[/i] if no two are parallel and no three are concurrent. Does there exist a value of $k$ such that given any $n$ lines in [i]standard form[/i], it is possible to colour the regions bounded by the $n$ lines using $k$ colours in such a way that no two regions of the same colour share a common intersection point of the $n$ lines?
1998 Mediterranean Mathematics Olympiad, 1
A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that
\[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]
2016 LMT, 9
An acute triangle has area $84$ and perimeter $42$, with each side being at least $10$ units long. Let $S$ be the set of points that are within $5$ units of some vertex of the triangle. What fraction of the area of $S$ lies outside the triangle?
[i]Proposed by Nathan Ramesh
2019 Singapore Senior Math Olympiad, 3
Let $a_1,a_2,\cdots,a_{2000}$ be distinct positive integers such that $1 \leq a_1 < a_2 < \cdots < a_{2000} < 4000$ such that the LCM (least common multiple) of any two of them is $\geq 4000$. Show that $a_1 \geq 1334$
1986 Canada National Olympiad, 4
For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$. Prove that $F(n,1)$ divides $F(n,k)$.
1978 Kurschak Competition, 1
$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.
1969 Canada National Olympiad, 7
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
PEN S Problems, 20
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a$, $b$, $c$ by \[a=\sqrt[3]{n}, \; b=\frac{1}{a-\lfloor a\rfloor}, \; c=\frac{1}{b-\lfloor b\rfloor}.\] Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra+sb+tc=0$.
2017 Hanoi Open Mathematics Competitions, 3
The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is
(A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.
BIMO 2022, 2
Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$.
What is the maximum possible value of $k$?
[i]Proposed by Ivan Chan Kai Chin[/i]
1982 IMO Shortlist, 17
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.