Found problems: 85335
2024 AMC 8 -, 4
When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. Which number did Yunji leave out?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2016 Harvard-MIT Mathematics Tournament, 10
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.
2013 India IMO Training Camp, 2
In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.
2019 Irish Math Olympiad, 1
De fine the [i]quasi-primes[/i] as follows.
$\bullet$ The first quasi-prime is $q_1 = 2$
$\bullet$ For $n \ge 2$, the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$.
Determine, with proof, whether or not $1000$ is a quasi-prime.
2006 Nordic, 2
Real numbers $x,y,z$ are not all equal and satisfy $x+\frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}=k$. Find all possible values of $k$.
2007 Nicolae Coculescu, 3
Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $ f(x)=\sin (x^2) , $ and let be a sequence $ \left( a_n \right)_{n\ge 0} $ with $ a_0\in (0,1) $ and defined as $ a_{n}=a_{n-1}-F\left( a_{n-1} \right) . $
Calculate $ \lim_{n\to\infty } a_n\sqrt{n} . $
[i]Florian Dumitrel[/i]
2023 USAMTS Problems, 2
Grogg takes an a × b × c rectangular block (where a, b, c are positive integers), paints
the outside of it purple, and cuts it into abc small 1 × 1 × 1 cubes. He then places all the
small cubes into a bag, and Winnie reaches in and randomly picks one of the small cubes.
If the probability that Winnie picks a totally unpainted cube is 20%, determine all possible
values of the number of cubes in the bag.
1978 AMC 12/AHSME, 30
In a tennis tournament, $n$ women and $2n$ men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $7/5$, then $n$ equals
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$
1989 IMO Shortlist, 5
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
2016 PUMaC Number Theory B, 2
For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$. If $n$ is a two-digit positive integer such that $\frac{n}{s(n)}$ is a multiple of $3$, compute the sum of all possible values of $n$.
2023 Oral Moscow Geometry Olympiad, 4
Given isosceles tetrahedron $PABC$ (faces are equal triangles). Let $A_0$, $B_0$ and $C_0$ be the touchpoints of the circle inscribed in the triangle $ABC$ with sides $BC$, $AC$ and $AB$ respectively, $A_1$, $B_1$ and $C_1$ are the touchpoints of the excircles of triangles $PCA$, $PAB$ and $PBC$ with extensions of sides $PA$, $PB$ and $PC$, respectively (beyond points $A$, $B$, $C$). Prove that the lines $A_0A_1$, $B_0B_1$ and $C_0C_1$ intersect at one point.
2006 Greece Junior Math Olympiad, 2
Find all positive integers $x , y$ which are roots of the equation
$2 x^y-y= 2005$
[u] Babis[/u]
2017 India PRMO, 29
For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$. For $n<100$, find the largest value of $h_n$.
2011 Tuymaada Olympiad, 2
Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Points $S_1$ and $S_2$ lie on the line $AB$ (but not between $A$ and $B$). The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$, and the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$. Prove that if the line $X_1X_2$ passes through $M$, then line $Y_1Y_2$ also passes through $M$.
1984 Czech And Slovak Olympiad IIIA, 4
Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.
2014 Harvard-MIT Mathematics Tournament, 4
Compute \[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\] (Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)
2008 AMC 10, 5
For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ x^2\plus{}y^2 \qquad
\textbf{(C)}\ 2x^2 \qquad
\textbf{(D)}\ 2y^2 \qquad
\textbf{(E)}\ 4xy$
2020 Thailand TST, 5
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
2022 AIME Problems, 9
Ellina has twelve blocks, two each of red $\left({\bf R}\right),$ blue $\left({\bf B}\right),$ yellow $\left({\bf Y}\right),$ green $\left({\bf G}\right),$ orange $\left({\bf O}\right),$ and purple $\left({\bf P}\right).$ Call an arrangement of blocks [i]even[/i] if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
$$ {\text {\bf R B B Y G G Y R O P P O}}$$is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1999 Mongolian Mathematical Olympiad, Problem 6
A point $M$ lies on the side $AC$ of a triangle $ABC$. The circle $\gamma$ with the diameter $BM$ intersects the lines $AB$ and $BC$ at $P$ and $Q$, respectively. Find the locus of the intersection point of the tangents to $\gamma$ at $P$ and $Q$ when point $M$ varies.
2016 Cono Sur Olympiad, 5
Let $ABC$ be a triangle inscribed on a circle with center $O$. Let $D$ and $E$ be points on the sides $AB$ and $BC$,respectively, such that $AD = DE = EC$. Let $X$ be the intersection of the angle bisectors of $\angle ADE$ and $\angle DEC$.
If $X \neq O$, show that, the lines $OX$ and $DE$ are perpendicular.
2010 LMT, 16
Determine the number of three digit integers that are equal to $19$ times the sum of its digits.
2024 Korea Winter Program Practice Test, Q3
Consider any sequence of real numbers $a_0$, $a_1$, $\cdots$. If, for all pairs of nonnegative integers $(m, s)$, there exists some integer $n \in [m+1, m+2024(s+1)]$ satisfying $a_m+a_{m+1}+\cdots+a_{m+s}=a_n+a_{n+1}+\cdots+a_{n+s}$, say that this sequence has [i]repeating sums[/i]. Is a sequence with repeating sums always eventually periodic?
1982 IMO Longlists, 55
Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.
2020 Sharygin Geometry Olympiad, 22
Let $\Omega$ be the circumcircle of cyclic quadrilateral $ABCD$. Consider such pairs of points $P$, $Q$ of diagonal $AC$ that the rays $BP$ and $BQ$ are symmetric with respect the bisector of angle $B$. Find the locus of circumcenters of triangles $PDQ$.