Found problems: 85335
2013 AMC 10, 24
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
$\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$
2021 IMC, 6
For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.
2018 Junior Balkan MO, 2
Find max number $n$ of numbers of three digits such that :
1. Each has digit sum $9$
2. No one contains digit $0$
3. Each $2$ have different unit digits
4. Each $2$ have different decimal digits
5. Each $2$ have different hundreds digits
2013 Korea National Olympiad, 5
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
2012 AMC 10, 6
The product of two positive numbers is $9$. The reciprocal of one of these numbers is $4$ times the reciprocal of the other number. What is the sum of the two numbers?
$ \textbf{(A)}\ \dfrac{10}{3}
\qquad\textbf{(B)}\ \dfrac{20}{3}
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ \dfrac{15}{2}
\qquad\textbf{(E)}\ 8
$
2006 Princeton University Math Competition, 5
Find the largest integer $k$ such that $12^k | 66!$.
LMT Speed Rounds, 15
Find the least positive integer $n$ greater than $1$ such that $n^3 -n^2$ is divisible by $7^2 \times 11$.
[i]Proposed by Jacob Xu[/i]
2009 Romanian Master of Mathematics, 1
For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$.
Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer.
[i]Dan Schwarz, Romania[/i]
1982 IMO Longlists, 14
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
PEN P Problems, 14
Let $n$ be a non-negative integer. Find all non-negative integers $a$, $b$, $c$, $d$ such that \[a^{2}+b^{2}+c^{2}+d^{2}= 7 \cdot 4^{n}.\]
2020 Taiwan TST Round 1, 2
Let point $H$ be the orthocenter of a scalene triangle $ABC$. Line $AH$ intersects with the circumcircle $\Omega$ of triangle $ABC$ again at point $P$. Line $BH, CH$ meets with $AC,AB$ at point $E$ and $F$, respectively. Let $PE, PF$ meet $\Omega$ again at point $Q,R$, respectively. Point $Y$ lies on $\Omega$ so that lines $AY,QR$ and $EF$ are concurrent. Prove that $PY$ bisects $EF$.
1983 Putnam, B5
Let $\lVert u\rVert$ denote the distance from the real number $u$ to the nearest integer. For positive integers $n$, let
$$a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.$$Determine $\lim_{n\to\infty}a_n$.
2010 Peru Iberoamerican Team Selection Test, P2
For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system.
Find all positive integers N for which there exist positive integers $a$,$b$,$c$, coprime two by two, such that:
$S(ab) = S(bc) = S(ca) = N$.
2000 JBMO ShortLists, 9
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2$.
2012 Turkey Team Selection Test, 2
In an acute triangle $ABC,$ let $D$ be a point on the side $BC.$ Let $M_1, M_2, M_3, M_4, M_5$ be the midpoints of the line segments $AD, AB, AC, BD, CD,$ respectively and $O_1, O_2, O_3, O_4$ be the circumcenters of triangles $ABD, ACD, M_1M_2M_4, M_1M_3M_5,$ respectively. If $S$ and $T$ are midpoints of the line segments $AO_1$ and $AO_2,$ respectively, prove that $SO_3O_4T$ is an isosceles trapezoid.
2020-2021 Winter SDPC, #4
Find all polynomials $P(x)$ with integer coefficients such that for all positive integers $n$, we have that $P(n)$ is not zero and $\frac{P(\overline{nn})}{P(n)}$ is an integer, where $\overline{nn}$ is the integer obtained upon concatenating $n$ with itself.
2009 All-Russian Olympiad, 1
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction).
2021-IMOC, A1
Find all real numbers x that satisfies$$\sqrt{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}+\sqrt{1-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}=x.$$
[url=https://artofproblemsolving.com/community/c6h2645263p22889979]2021 IMOC Problems[/url]
2017 AMC 10, 17
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^2+y^2=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS }$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 3\sqrt{5}\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 5\sqrt{2}$
2007 ITAMO, 3
Let ABC be a triangle, G its centroid, M the midpoint of AB, D the point on the line $AG$ such that $AG = GD, A \neq D$, E the point on the line $BG$ such that $BG = GE, B \neq E$. Show that the quadrilateral BDCM is cyclic if and only if $AD = BE$.
1986 Canada National Olympiad, 2
A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers. The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?
2023 Belarus Team Selection Test, 3.3
Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.
2021 AMC 12/AHSME Fall, 23
A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$
$\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$
Kyiv City MO 1984-93 - geometry, 1990.10.5
A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of the quadrilateral $ABCD$.
2012 AIME Problems, 10
Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n = x \lfloor x \rfloor$.
[b]Note[/b]: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.