Found problems: 85335
2012 Today's Calculation Of Integral, 804
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$
2022 Harvard-MIT Mathematics Tournament, 5
Let $ABC$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $BC$ such that $BE = EF = F C$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, so that $X$, $Y$ , and $G$ are not collinear. If the line through $E$ parallel to $XG$ and the line through $F$ parallel to $Y G$ intersect at $P\ne G$, prove that $GP$ passes through the midpoint of $XY$.
2023 SAFEST Olympiad, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
2005 Bosnia and Herzegovina Junior BMO TST, 2
Let n be a positive integer. Prove the following statement:
”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”
2002 Tournament Of Towns, 5
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?
2008 AMC 12/AHSME, 25
Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$?
$ \textbf{(A)}\ 28\sqrt{3}\qquad
\textbf{(B)}\ 30\sqrt{3}\qquad
\textbf{(C)}\ 32\sqrt{3}\qquad
\textbf{(D)}\ 35\sqrt{3}\qquad
\textbf{(E)}\ 36\sqrt{3}$
2005 Thailand Mathematical Olympiad, 7
How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?
2021 Indonesia TST, N
Let $n$ be a positive integer. Prove that $$\gcd(\underbrace{11\dots 1}_{n \text{times}},n)\mid 1+10^k+10^{2k}+\dots+10^{(n-1)k}$$ for all positive integer $k$.
2013 Dutch IMO TST, 1
Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\
bc + d + a = 5 \\
cd + a + b = 2 \\
da + b + c = 6 \end{cases}$
2023 Korea Summer Program Practice Test, P6
$AB < AC$ on $\triangle ABC$. The midpoint of arc $BC$ which doesn't include $A$ is $T$ and which includes $A$ is $S$. On segment $AB,AC$, $D,E$ exist so that $DE$ and $BC$ are parallel. The outer angle bisector of $\angle ABE$ and $\angle ACD$ meets $AS$ at $P$ and $Q$. Prove that the circumcircle of $\triangle PBE$ and $\triangle QCD$ meets on $AT$.
2025 Al-Khwarizmi IJMO, 1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}
and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.
[i]Marek Maruin, Slovakia [/i]
2018 PUMaC Algebra A, 1
Let
$$a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1$$
The value of $\sum_{k = 1}^\infty a_k$ can be expressed as a rational number $\frac{p}{q}$ in simplest form. Find $p + q$.
2018 Turkey MO (2nd Round), 3
A sequence $a_1,a_2,\dots$ satisfy
$$
\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},
$$
for every $n\in\mathbb{N}$.
Let $c$ be a positive integer. Prove that, for every positive integer $n$,
$$
\frac{c^{a_n}-c^{a_{n-1}}}{n}
$$
is an integer.
2004 All-Russian Olympiad Regional Round, 10.4
$N \ge 3$ different points are marked on the plane. It is known that among pairwise distances between marked points there are not more than $n$ different distances. Prove that $N \le (n + 1)^2$.
2019 India IMO Training Camp, P2
Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$.
Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$.
[i]Proposed by Anant Mudgal[/i]
2025 Poland - First Round, 12
We will say that a subset $A$ of the set of non-negative integers is $cool$, if there exist an integer $k$, such that for every integer $n\geq k$ there exists exactly one pair of integers $a>b$ from $A$ such that $n=a+b$. Decide, if there exists a $cool$ set.
2020 Online Math Open Problems, 7
On a $5 \times 5$ grid we randomly place two \emph{cars}, which each occupy a single cell and randomly face in one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequence of moves such that, afterward, both cars occupy the same cell is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
[i]Proposed by Sean Li[/i]
1969 Miklós Schweitzer, 7
Prove that if a sequence of Mikusinski operators of the form $ \mu e^{\minus{}\lambda s}$ ( $ \lambda$ and $ \mu$ nonnegative real
numbers, $ s$ the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form.
[i]E. Geaztelyi[/i]
2000 239 Open Mathematical Olympiad, 3
For all positive real numbers $a_1, a_2, \dots, a_n$, prove that
$$
\frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot
\frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot
\frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot
\frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.$$
2014 ASDAN Math Tournament, 4
If Bobby’s age is increased by $6$, it’s a number with an integral (positive) square root. If his age is decreased by $6$, it’s that square root. How old is Bobby?
2016 Postal Coaching, 3
Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.
2024 Austrian MO National Competition, 2
Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$.
[i](Karl Czakler)[/i]
2014-2015 SDML (High School), 2
A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?
2002 AMC 12/AHSME, 18
If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$.
$\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$
2002 Junior Balkan MO, 2
Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$.
[i]Ciprus[/i]