Found problems: 85335
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 $x$.
2020 MIG, 11
The numbers $1$, $2$, $3$, $4$, $5$, $6$ are placed onto the following six spots such that the average of the leftmost two spots, middle two spots, and rightmost two spots are all equal. What is the difference between the largest and smallest possibilities of the number on the shaded spot shown below?
[asy]
size(110);
draw(Circle((0,0),0.7));
draw(Circle((2,0),0.7));label("$1$",(2,0));
filldraw(Circle((4,0),0.7),gray);
draw(Circle((6,0),0.7));
draw(Circle((8,0),0.7));
draw(Circle((10,0),0.7));label("$2$",(10,0));
[/asy]
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2022 Taiwan TST Round 1, 4
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
1999 Tournament Of Towns, 1
A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?
(Tairova)
2021 MOAA, 22
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
[i]Proposed by Andy Xu[/i]
2012 Centers of Excellency of Suceava, 3
Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ that has a root, and for which the line $ y=0 $ in the Cartesian plane is an horizontal asymptote. Show that $ f $ is bounded and touches its boundaries.
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]
2012 JBMO ShortLists, 4
Solve the following equation for $x , y , z \in \mathbb{N}$ :
\[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]
2016 Germany Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2011 JBMO Shortlist, 3
Find all positive integers $n$ such that the equation $y^2 + xy + 3x = n(x^2 + xy + 3y)$ has at least a solution $(x, y)$ in positive integers.
1993 India National Olympiad, 5
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.
2024 CIIM, 5
A board of size $3 \times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Prove that
\[
\lim_{N \to \infty} \frac{a(N)}{N}
\]
exists and determine its value.
2002 Flanders Junior Olympiad, 4
Two congruent right-angled isosceles triangles (with baselength 1) slide on a line as on the picture. What is the maximal area of overlap?
[img]https://cdn.artofproblemsolving.com/attachments/a/8/807bb5b760caaa600f0bac95358963a902b1e7.png[/img]