Found problems: 85335
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]
LMT Speed Rounds, 2010.8
How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$
2017 Saint Petersburg Mathematical Olympiad, 4
A positive integer $n$ is called almost-square if $n$ can be represented as $n=ab$ where $a,b$ are positive integers that $a\leq b\leq 1.01a$. Prove that there exists infinitely many positive integers $m$ that there’re no almost-square positive integer among $m,m+1,…,m+198$.
2005 Junior Balkan Team Selection Tests - Moldova, 4
Let the $A$ be the set of all nonenagative integers.
It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds:
[b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$;
[b]2)[/b] $f(2n) < 6f(n)$,
Find all solutions of $f(k)+f(l) = 293$, $k<l$.
2012 Uzbekistan National Olympiad, 3
The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.
2018 Harvard-MIT Mathematics Tournament, 4
Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.
2005 Gheorghe Vranceanu, 1
Let be a natural number $ n\ge 2 $ and the $ n\times n $ matrix whose entries at the $ \text{i-th} $ line and $ \text{j-th} $ column is $ \min (i,j) . $ Calculate:
[b]a)[/b] its determinant.
[b]b)[/b] its inverse.
2007 Nicolae Păun, 4
Prove that for any natural number $ n, $ there exists a number having $ n+1 $ decimal digits, namely, $ k_0,k_1,k_2,\ldots ,k_n $, and a $ \text{(n+1)-tuple}, $ say $\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , $ that satisfies:
$$ 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2} $$
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
2014 Singapore Junior Math Olympiad, 5
In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.