Found problems: 85335
2022 AIME Problems, 4
Let $w = \frac{\sqrt{3}+i}{2}$ and $z=\frac{-1+i\sqrt{3}}{2}$, where $i=\sqrt{-1}$. Find the number of ordered pairs $(r, s)$ of positive integers not exceeding $100$ that satisfy the equation $i\cdot w^r=z^s$.
2011 AMC 10, 4
Let $X$ and $Y$ be the following sums of arithmetic sequences: \begin{eqnarray*} X &=& 10 + 12 + 14 + \cdots + 100, \\ Y &=& 12 + 14 + 16 + \cdots + 102. \end{eqnarray*} What is the value of $Y - X$?
$ \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 $
1984 Spain Mathematical Olympiad, 7
Consider the natural numbers written in the decimal system.
(a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have?
(b) Prove that there is no number that decreases $12$ times when its first digit is erased.
(c) Find the necessary and sufficient condition on $k$ for the existence of a natural number which is divided by $k$ when its first digit is erased.
2000 Brazil Team Selection Test, Problem 1
Show that if the sides $a, b, c$ of a triangle satisfy the equation
\[2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,\]
then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.
2019 LIMIT Category B, Problem 12
Find the number of rational solutions of the following equations (i.e., rational $x$ and $y$ satisfy the equations)
$$x^2+y^2=2$$$$x^2+y^2=3$$$\textbf{(A)}~2\text{ and }2$
$\textbf{(B)}~2\text{ and }0$
$\textbf{(C)}~2\text{ and infinitely many}$
$\textbf{(D)}~\text{Infinitely many and }0$
1998 AMC 8, 8
A child's wading pool contains $200$ gallons of water. If water evaporates at the rate of $0.5$ gallons per day and no other water is added or removed, how many gallons of water will be in the pool after $30$ days?
$ \text{(A)}\ 140\qquad\text{(B)}\ 170\qquad\text{(C)}\ 185\qquad\text{(D)}\ 198.5\qquad\text{(E)}\ 199.85 $
1992 AMC 12/AHSME, 23
What is the size of the largest subset, $S$, of $\{1, 2, 3, \ldots, 50\}$ such that no pair of distinct elements of $S$ has a sum divisible by $7$?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 23 $
2025 Kyiv City MO Round 2, Problem 1
Mykhailo chose three distinct real numbers \( a, b, c \) and wrote the following numbers on the board:
\[
a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca.
\]What is the minimum possible number of distinct numbers that can be written on the board?
[i]Proposed by Anton Trygub[/i]
1996 AMC 12/AHSME, 9
Triangle $PAB$ and square $ABCD$ are in perpendicular planes. Given that $PA = 3, PB = 4,$ and $AB = 5$, what is $PD$?
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ \sqrt{34} \qquad
\textbf{(C)}\ \sqrt{41} \qquad
\textbf{(D)}\ 2\sqrt{13} \qquad
\textbf{(E)}\ 8$
1996 Estonia National Olympiad, 3
There are $1,000,000$ piles of $1996$ coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs $9$ grams, and each real coin weighs $10$ grams?
2002 Vietnam Team Selection Test, 3
Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1 $. Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions:
[b]I.[/b] $m\leq x\leq n$ for all $x\in S$;
[b]II.[/b] the product of all elements in $S$ is the square of an integer.
2019 Puerto Rico Team Selection Test, 4
Rectangle $ABCD$ has sides $AB = 3$, $BC = 2$. Point $ P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$. Find $BP$.
2007 National Olympiad First Round, 15
What is the minimum value of $ab+cd$, if $ab+cd = ef+gh$ where $a,b,c,d,e,f,g,h$ are distinct positive integers?
$
\textbf{(A)}\ 34
\qquad\textbf{(B)}\ 33
\qquad\textbf{(C)}\ 32
\qquad\textbf{(D)}\ 31
\qquad\textbf{(E)}\ 30
$
1990 IMO Shortlist, 7
Let $ f(0) \equal{} f(1) \equal{} 0$ and
\[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\]
Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$
2016 Iranian Geometry Olympiad, 4
Let $\omega$ be the circumcircle of right-angled triangle $ABC$ ($\angle A = 90^{\circ}$). The tangent to $\omega$ at point $A$ intersects the line $BC$ at point $P$. Suppose that $M$ is the midpoint of the minor arc $AB$, and $PM$ intersects $\omega$ for the second time in $Q$. The tangent to $\omega$ at point $Q$ intersects $AC$ at $K$. Prove that $\angle PKC = 90^{\circ}$.
[i]Proposed by Davood Vakili[/i]
1997 Estonia National Olympiad, 2
Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.
1973 Canada National Olympiad, 4
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.
[img]6740[/img]
2018 Putnam, A1
Find all ordered pairs $(a, b)$ of positive integers for which
\[\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.\]
2019 Online Math Open Problems, 16
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that \[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\] If \[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\] then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Tristan Shin[/i]
2013 Balkan MO Shortlist, C2
Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.
2007 Pre-Preparation Course Examination, 19
Find all functions $f : \mathbb N \to \mathbb N$ such that:
i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$,
ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and
iii) $f(m)=1$ if and only if $m=1$.
2012 Tournament of Towns, 4
A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.
2022 Spain Mathematical Olympiad, 6
Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]
2020 Malaysia IMONST 1, 13
Given a right-angled triangle with perimeter $18$. The sum of the squares
of the three side lengths is $128$. What is the area of the triangle?
2006 Purple Comet Problems, 4
At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does the coo-coo bird coo-coo from $12:42$ PM on Monday until $3:42$ AM on Wednesday?