Found problems: 85335
1989 Polish MO Finals, 2
Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.
Durer Math Competition CD 1st Round - geometry, 2013.C1
Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle?
[img]https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667caafee6dbf397632e57e0.png[/img]
1971 Putnam, A2
Determine all polynomials $P(x)$ such that $P(x^2+1)=(P(x))^2+1$ and $P(0)=0.$
2002 Swedish Mathematical Competition, 5
The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.
2002 AMC 12/AHSME, 18
A point $ P$ is randomly selected from the rectangular region with vertices $ (0, 0)$, $ (2, 0)$, $ (2, 1)$, $ (0, 1)$. What is the probability that $ P$ is closer to the origin than it is to the point $ (3, 1)$?
$ \textbf{(A)}\ \frac{1}{2} \qquad
\textbf{(B)}\ \frac{2}{3} \qquad
\textbf{(C)}\ \frac{3}{4} \qquad
\textbf{(D)}\ \frac{4}{5} \qquad
\textbf{(E)}\ 1$
2024 All-Russian Olympiad Regional Round, 11.4
Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.
2016 Azerbaijan Junior Mathematical Olympiad, 3
$65$ distinct natural numbers not exceeding $2016$ are given. Prove that among these numbers we can find four $a,b,c,d$ such that $a+b-c-d$ is divisible by $2016.$
2022 Korea Junior Math Olympiad, 2
For positive integer $n \ge 3$, find the number of ordered pairs $(a_1, a_2, ... , a_n)$ of integers that satisfy the following two conditions
[list=disc]
[*]For positive integer $i$ such that $1\le i \le n$, $1 \le a_i \le i$
[*]For positive integers $i,j,k$ such that $1\le i < j < k \le n$, if $a_i = a_j$ then $a_j \ge a_k$
[/list]
1979 IMO Longlists, 13
The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let $M$ be an arbitrary set of $n$ squares of this net. Prove that it is possible to choose no fewer than $n/4$ squares of $M$ in such a way that no two of them have a common point.
2021 Czech and Slovak Olympiad III A, 4
Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$).
(Tomáš Bárta)
2018 Saudi Arabia JBMO TST, 1
Let $n$ be a natural composite number. For each proper divisor $d$ of $n$ we write the number $d + 1$ on the board. Determine all natural numbers $n$ for which the numbers written on the board are all the proper divisors of a natural number $m$. (The proper divisors of a natural number $a> 1$ are the positive divisors of $a$ different from $1$ and $a$.)
1953 AMC 12/AHSME, 44
In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $ 8$ and $ 2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $ \minus{}9$ and $ \minus{}1$ for the roots. The correct equation was:
$ \textbf{(A)}\ x^2\minus{}10x\plus{}9\equal{}0 \qquad\textbf{(B)}\ x^2\plus{}10x\plus{}9\equal{}0 \qquad\textbf{(C)}\ x^2\minus{}10x\plus{}16\equal{}0\\
\textbf{(D)}\ x^2\minus{}8x\minus{}9\equal{}0 \qquad\textbf{(E)}\ \text{none of these}$
2018 Junior Regional Olympiad - FBH, 1
Price of some item has decreased by $5\%$. Then price increased by $40\%$ and now it is $1352.06\$$ cheaper than doubled original price. How much did the item originally cost?
1992 India National Olympiad, 4
Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.
2002 Chile National Olympiad, 3
Given the line $AB$, let $M$ be a point on it. Towards the same side of the plane and with bases $AM$ and $MB$, squares $AMCD$ and $MBEF$ are constructed. Let $N$ be the point (different from $M$) where the circumcircles circumscribed to both squares intersect and let $N_1$ be the point where the lines $BC$ and $AF$ intersect. Prove that the points $N$ and $N_1$ coincide. Prove that as the point $M$ moves on the line $AB$, the line $MN$ moves always passing through a fixed point.
1997 Romania National Olympiad, 4
The quadrilateral $ABCD$ has two parallel sides. Let $M$ and $N$ be the midpoints of $[DC]$ and $[BC]$, and $P$ the common point of the lines $AM$ and $DN$. If $\frac{PM}{AP}=\frac{1}{4}$, prove that $ABCD$ is a parallelogram.
1993 IberoAmerican, 3
Two nonnegative integers $a$ and $b$ are [i]tuanis[/i] if the decimal expression of $a+b$ contains only $0$ and $1$ as digits. Let $A$ and $B$ be two infinite sets of non negative integers such that $B$ is the set of all the [i]tuanis[/i] numbers to elements of the set $A$ and $A$ the set of all the [i]tuanis[/i] numbers to elements of the set $B$. Show that in at least one of the sets $A$ and $B$ there is an infinite number of pairs $(x,y)$ such that $x-y=1$.
2007 Hong Kong TST, 2
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 2
Let $A$, $B$ and $C$ be real numbers such that
(i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$,
(ii) $\tan C$ and $\cot C$ are defined.
Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.
2022 Middle European Mathematical Olympiad, 2
Let $n$ be a positive integer. Anna and Beatrice play a game with a deck of $n$ cards labelled with the numbers $1, 2,...,n$. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if $k$ denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the $k$ topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.
2023 Math Prize for Girls Problems, 15
A square is divided into four non-overlapping isosceles triangles. Let $X$ be the degree measure of one of the twelve angles of these four triangles. Compute the sum of all possible different values of $X$. (Consider all possible diagrams.)
2004 AIME Problems, 14
A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is $\frac{a-\sqrt{b}}c$ feet, where $a, b,$ and $c$ are positive integers, and $c$ is prime. Find $a+b+c$.
2015 Benelux, 3
Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?
2017-IMOC, G3
Let $ABCD$ be a circumscribed quadrilateral with center $O$. Assume the incenters of $\vartriangle AOC, \vartriangle BOD$ are $I_1, I_2$, respectively. If circumcircles of $\vartriangle AI_1C$ and $\vartriangle BI_2D$ intersect at $X$, prove the following identity:
$(AB \cdot CX \cdot DX)^2 + (CD\cdot AX \cdot BX)^2 = (AD\cdot BX \cdot CX)^2 + (BC \cdot AX \cdot DX)^2$
2000 South africa National Olympiad, 4
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.
2010 Contests, 2
A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for
some integer $\ell \ge 0$.
(a) Show that no number is both polite and a power of two.
(b) Show that every positive integer is polite or a power of two.