Found problems: 85335
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 $
2025 Harvard-MIT Mathematics Tournament, 9
Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$
$\quad \bullet \ f(x+p)=f(x)\text{ and}$
$\quad \bullet \ p \text{ divides } f(x+f(x))-x.$
2008 Saint Petersburg Mathematical Olympiad, 1
We color some cells in $10000 \times 10000$ square, such that every $10 \times 10$ square and every $1 \times 100$ line have at least one coloring cell. What minimum number of cells we should color ?
1999 All-Russian Olympiad Regional Round, 10.5
Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?
2018 HMNT, 2
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?
2006 All-Russian Olympiad Regional Round, 8.6
In a checkered square $101 \times 101$, each cell of the inner square $99 \times 99$ is painted in one of ten colors (cells adjacent to the border of the square, not painted). Could it turn out that in every in a $3\times 3$ square, is exactly one more cell painted the same color as the central cell?