Found problems: 85335
1974 IMO Longlists, 29
Let $A,B,C,D$ be points in space. If for every point $M$ on the segment $AB$ the sum
\[S_{AMC}+S_{CMD}+S_{DMB}\]
Is constant show that the points $A,B,C,D$ lie in the same plane.
[hide="Note."]
[i]Note. $S_X$ denotes the area of triangle $X.$[/i][/hide]
2018 Brazil Undergrad MO, 6
Given an equilateral triangle $ABC$ in the plane, how many points $P$ in the plane are such that the three triangles $AP B, BP C $ and $CP A$ are isosceles and not degenerate?
2015 Moldova Team Selection Test, 2
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove the following inequality: \\$a^3+b^3+c^3+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2} \geq \frac{9}{2}$.
2017 ASDAN Math Tournament, 3
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$. Find $ab$.
2010 AIME Problems, 3
Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.
1993 Greece National Olympiad, 7
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
2016 Turkmenistan Regional Math Olympiad, Problem 4
Let $ABC$ is isosceles triangle $AB=AC$. The point $P$ inside $ABC$ triangle such that angle $\widehat{BCP}=30^o$ , $\widehat{APB}=150^o$ and $\widehat{CAP}=39^o$ . Find $\widehat{BAP}$
2004 Harvard-MIT Mathematics Tournament, 5
A rectangle has perimeter $10$ and diagonal $\sqrt{15}$. What is its area?
2012 Rioplatense Mathematical Olympiad, Level 3, 1
An integer $n$ is called [i]apocalyptic[/i] if the addition of $6$ different positive divisors of $n$ gives $3528$. For example, $2012$ is apocalyptic, because it has six divisors, $1$, $2$, $4$, $503$, $1006$ and $2012$, that add up to $3528$.
Find the smallest positive apocalyptic number.
2014 Indonesia MO, 2
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
2000 AIME Problems, 1
Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0.$
2020 Spain Mathematical Olympiad, 6
Let $S$ be a finite set of integers. We define $d_2(S)$ and $d_3(S)$ as:
$\bullet$ $d_2(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^2-y^2 = a$
$\bullet$ $d_3(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^3-y^3 = a$
(a) Let $m$ be an integer and $S = \{m, m+1, \ldots, m+2019\}$. Prove:
$$d_2(S) > \frac{13}{7} d_3(S)$$
(b) Let $S_n = \{1, 2, \ldots, n\}$ with $n$ a positive integer. Prove that there exists a $N$ so that for all $n > N$:
$$ d_2(S_n) > 4 \cdot d_3(S_n) $$
2019 LIMIT Category B, Problem 8
Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$?
2004 Bosnia and Herzegovina Team Selection Test, 2
Determine whether does exists a triangle with area $2004$ with his sides positive integers.
2013 Sharygin Geometry Olympiad, 6
The altitudes $AA_1, BB_1, CC_1$ of an acute triangle $ABC$ concur at $H$. The perpendicular lines from $H$ to $B_1C_1, A_1C_1$ meet rays $CA, CB$ at $P, Q$ respectively. Prove that the line from $C$ perpendicular to $A_1B_1$ passes through the midpoint of $PQ$.
1960 AMC 12/AHSME, 40
Given right triangle $ABC$ with legs $BC=3$, $AC=4$. Find the length of the shorter [i]angle trisector[/i] from $C$ to the hypotenuse:
$ \textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad\textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad\textbf{(C)}\ 6\sqrt{3}-8\qquad\textbf{(D)}\ \frac{5\sqrt{10}}{6} \qquad$
$\textbf{(E)}\ \frac{25}{12}$
2021 USEMO, 2
Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors.
[i]Proposed by Ankan Bhattacharya [/i]
2020 Princeton University Math Competition, B1
The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.
2015 Latvia Baltic Way TST, 16
Points $X$ , $Y$, $Z$ lie on a line $k$ in this order. Let $\omega_1$, $\omega_2$, $\omega_3$ be three circles of diameters $XZ$, $XY$ , $YZ$ , respectively. Line $\ell$ passing through point $Y$ intersects $\omega_1$ at points $A$ and $D$, $\omega_2$ at $B$ and $\omega_3$ at $C$ in such manner that points $A, B, Y, X, D$ lie on $\ell$ in this order. Prove that $AB =CD$.
2010 Ukraine Team Selection Test, 10
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
2018 Yasinsky Geometry Olympiad, 5
The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?
2022 Bulgaria JBMO TST, 2
Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.
2019 AMC 10, 13
Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$
$\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$
2007 Princeton University Math Competition, 6
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
2012 AMC 8, 23
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon?
$\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}4\sqrt3 \qquad \textbf{(E)}\hspace{.05in}6\sqrt3 $