Found problems: 85335
2019 Dutch BxMO TST, 2
Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$
2017 ITAMO, 5
Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.
2021 Malaysia IMONST 1, 16
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?
1966 IMO Shortlist, 48
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
2010 China Team Selection Test, 2
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
1954 AMC 12/AHSME, 32
The factors of $ x^4\plus{}64$ are:
$ \textbf{(A)}\ (x^2\plus{}8)^2 \qquad
\textbf{(B)}\ (x^2\plus{}8)(x^2\minus{}8) \qquad
\textbf{(C)}\ (x^2\plus{}2x\plus{}4)(x^2\minus{}8x\plus{}16) \\
\textbf{(D)}\ (x^2\minus{}4x\plus{}8)(x^2\minus{}4x\minus{}8) \qquad
\textbf{(E)}\ (x^2\minus{}4x\plus{}8)(x^2\plus{}4x\plus{}8)$
2004 Alexandru Myller, 3
Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.
1979 Czech And Slovak Olympiad IIIA, 4
Let $n$ be any natural number. Find all $n$-tuples of real numbers $x_1\le x_2\le ... \le x_n$, for which holds
$$\left(\sum_{i=1}^n x_i\right)^2 \le n \sum_{i=1}^n x_i x_{n-i+1}.$$
1959 AMC 12/AHSME, 27
Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$?
$ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$
$\textbf{(B)}\ \text{The discriminant is 9}\qquad$
$\textbf{(C)}\ \text{The roots are imaginary}\qquad$
$\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$
$\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $
2008 Stanford Mathematics Tournament, 15
While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at $ 10 \text{m/s}$, and you manage a three-second head start over the raptor. If the raptor runs at $ 15\sqrt{2} \text{m/s}$, but only runs either north or east at any given time, how many seconds do you have until it devours you?
2006 QEDMO 2nd, 2
There are $N$ cities in the country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.
1993 Tournament Of Towns, (374) 2
A square is constructed on the side $AB$ of triangle $ABC$ (outside the triangle).$ O$ is the centre of the square. $M$ and $N$ are the midpoints of the sides $BC$ and $AC$. The lengths of these sides are $a$ and $b$ respectively. Find the maximal possible value of the sum $CM + ON$ (when the angle at $C$ changes).
(IF Sharygin)
2004 All-Russian Olympiad Regional Round, 9.4
Three natural numbers are such that the product of any two of them is divided by the sum of these two numbers. Prove that these three numbers have a common divisor greater than one.
2024 Junior Balkan Team Selection Tests - Romania, P3
In the exterior of the acute-angles triangle $ABC$ we construct the isosceles triangles $DAB$ and $EAC$ with bases $AB{}$ and $AC{}$ respectively such that $\angle DBC=\angle ECB=90^\circ.$ Let $M$ and $N$ be the reflections of $A$ with respect to $D$ and $E$ respectively. Prove that the line $MN$ passes through the orthocentre of the triangle $ABC.$
[i]Florin Bojor[/i]
2002 AMC 10, 11
The product of three consecutive positive integers is $ 8$ times their sum. What is the sum of their squares?
$ \textbf{(A)}\ 50 \qquad
\textbf{(B)}\ 77 \qquad
\textbf{(C)}\ 110 \qquad
\textbf{(D)}\ 149 \qquad
\textbf{(E)}\ 194$
2014 Online Math Open Problems, 11
Given a triangle $ABC$, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $BE=1$, $EF=24$, and $FC=3$, find the perimeter of $\triangle{ABC}$.
[i]Proposed by Ray Li[/i]
2020-21 IOQM India, 12
Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$, how many chords can be drawn before returning to the starting point ?
[img]https://i.imgur.com/Cg37vwa.png[/img]
2022 HMNT, 3
Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.
2009 Ukraine National Mathematical Olympiad, 4
In the triangle $ABC$ given that $\angle ABC = 120^\circ .$ The bisector of $\angle B$ meet $AC$ at $M$ and external bisector of $\angle BCA$ meet $AB$ at $P.$ Segments $MP$ and $BC$ intersects at $K$. Prove that $\angle AKM = \angle KPC .$
2024 USAMO, 6
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$. A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$-large if $|A_i| \ge \ell$ for all $1 \le i \le k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality
\[ \sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2 \]
holds for all positive integer $k$, all nonnegative real numbers $x_1,x_2,\dots,x_k$, and all $\ell$-large collections $A_1,A_2,\dots,A_k$ of subsets of $\{1,2,\dots,n\}$.
[i]Proposed by Titu Andreescu and Gabriel Dospinescu[/i]
1973 Spain Mathematical Olympiad, 1
Given the sequence $(a_n)$, in which $a_n =\frac14 n^4 - 10n^2(n - 1)$, with $n = 0, 1, 2,...$ Determine the smallest term of the sequence.
2001 Tournament Of Towns, 5
Let $a$ and $d$ be positive integers. For any positive integer $n$, the number $a+nd$ contains a block of consecutive digits which constitute the number $n$. Prove that $d$ is a power of 10.
Durer Math Competition CD 1st Round - geometry, 2009.C3
We know the lengths of the $3$ altitudes of a triangle. Construct the triangle.
2005 China Team Selection Test, 3
Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.
2001 National High School Mathematics League, 3
In four functions $y=\sin|x|,y=\cos|x|,y=|\cot x|,y=\lg|\sin x|$, which one is even function, and increases on $\left(0,\frac{\pi}{2}\right)$, with period of $\pi$?
$\text{(A)}y=\sin|x|\qquad\text{(B)}y=\cos|x|\qquad\text{(C)}y=|\cot x|\qquad\text{(D)}y=\lg|\sin x|$