Found problems: 85335
2018 CMIMC Algebra, 4
2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero. At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right. Suppose the new numbers on the ducklings sum to 1000. What is the maximum possible sum of the original numbers on all 2018 slips?
2024 Miklos Schweitzer, 5
Let $X$ be a regular topological space and let $S$ be a countably compact dense subspace in $X$. (The countably compact property means that every infinite subset of $S$ has an accumulation point in $S$.) Show that $S$ is also $G_\delta$-dense in $X$, i.e., $S$ intersects all nonempty $G_\delta$ sets.
PEN I Problems, 13
Suppose that $n \ge 2$. Prove that \[\sum_{k=2}^{n}\left\lfloor \frac{n^{2}}{k}\right\rfloor = \sum_{k=n+1}^{n^{2}}\left\lfloor \frac{n^{2}}{k}\right\rfloor.\]
2020 Miklós Schweitzer, 11
Given a real number $p>1$, a continuous function $h\colon [0,\infty)\to [0,\infty)$, and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$, prove the following inequality
\[\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.\]
I Soros Olympiad 1994-95 (Rus + Ukr), 9.4
Use a compass and a ruler to construct a triangle, given the intersection point of its median, the orthocenter, and one from the vertices.
2018 International Olympic Revenge, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that
\[
f(x)^2-f(y)^2=f(x+y)\cdot f(x-y),
\]
for all $x,y\in \mathbb{Q}$.
[i]Proposed by Portugal.[/i]
MOAA Accuracy Rounds, 2021.7
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
2017 QEDMO 15th, 10
Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .
2013 Online Math Open Problems, 24
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$.
[i]Ray Li[/i]
2011 Bogdan Stan, 3
Find all Riemann integrable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ which have the property that, for all nonconstant and continuous functions $ g:\mathbb{R}\longrightarrow\mathbb{R}, $ and all real numbers $ a,b $ such that $ a<b, $ the following equality holds.
$$ \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx $$
[i]Cosmin Nițu[/i]
2023 CMIMC Algebra/NT, 7
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$.
[i]Proposed by Giacomo Rizzo[/i]
2019 District Olympiad, 1
Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$
2005 Greece Junior Math Olympiad, 1
We are given a trapezoid $ABCD$ with $AB \parallel CD$, $CD=2AB$ and $DB \perp BC$. Let $E$ be the intersection of lines $DA$ and $CB$, and $M$ be the midpoint of $DC$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that triangle $CDE$ is isosceles.
(c) If $AM$ and $BD$ meet at $O$, and $OE$ and $AB$ meet at $N,$ prove that the line $DN$ bisects segment $EB$.
2007 Baltic Way, 19
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$.
A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$.
Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.
2019 India PRMO, 22
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
2013 Princeton University Math Competition, 11
If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?
1997 Denmark MO - Mohr Contest, 1
Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?
2018 ELMO Shortlist, 4
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.
[i]Proposed by Carl Schildkraut[/i]
1995 AMC 12/AHSME, 6
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$?
[asy]
defaultpen(linewidth(0.7));
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);
draw(shift(1,0)*p, dashed);
label("$x$", (0.3,0.5), E);
label("$A$", (1.3,0.5), E);
label("$B$", (1.3,1.5), E);
label("$C$", (2.3,1.5), E);
label("$D$", (2.3,2.5), E);
label("$E$", (3.3,2.5), E);[/asy]
$
\mathbf{(A)}\; A\qquad
\mathbf{(B)}\; B\qquad
\mathbf{(C)}\; C\qquad
\mathbf{(D)}\; D\qquad
\mathbf{(E)}\; E$
2021 Serbia National Math Olympiad, 3
In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.
2011 China Second Round Olympiad, 2
Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.
1977 Putnam, B4
Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exists points $P_1$ and $P_2$ on $C$ such that $Q$ is the midpoint of the line segment $P_1P_2.$
2005 Today's Calculation Of Integral, 35
Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.
2024 AMC 8 -, 5
Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. Which of the following integers [i]cannot[/i] be the sum of the two numbers?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2007 Nicolae Coculescu, 4
Let be three nonnegative integers $ m,n,p $ and three real numbers $ x,y,z $ such that $ 2^mx+2^ny+2^pz\ge 0. $ Prove:
$$ 2^m\left( 2^x-1 \right)+2^n\left( 2^y-1 \right)+2^p\left( 2^z-1 \right)\ge 0 $$
[i]Cristinel Mortici[/i]