Found problems: 85335
1999 Harvard-MIT Mathematics Tournament, 2
A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?
2021 China Second Round Olympiad, Problem 5
Define the regions $M, N$ in the Cartesian Plane as follows:
\begin{align*}
M &= \{(x, y) \in \mathbb R^2 \mid 0 \leq y \leq \text{min}(2x, 3-x)\} \\
N &= \{(x, y) \in \mathbb R^2 \mid t \leq x \leq t+2 \}
\end{align*}
for some real number $t$. Denote the common area of $M$ and $N$ for some $t$ be $f(t)$. Compute the algebraic form of the function $f(t)$ for $0 \leq t \leq 1$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 5)[/i]
Kvant 2023, M2771
For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only?
Alexey Glebov
1993 Spain Mathematical Olympiad, 2
In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above.
$0 \, 1\, 2\, 3 \,4\, ... \,1991 \,1992\, 1993$
$\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985$
$\,\,\,4 \,8 \,12\, .......... \,\,\,7968$
·······································
Prove that the bottom number is a multiple of $1993$.
2014 South africa National Olympiad, 3
In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.
2005 Postal Coaching, 16
The diagonals AC and BD of a cyclic ABCD intersect at E. Let O be circumcentre of ABCD. If midpoints of AB, CD, OE are collinear prove that AD=BC.
Bomb
[color=red][Moderator edit: The problem is wrong. See also http://www.mathlinks.ro/Forum/viewtopic.php?t=53090 .][/color]
2023 Sharygin Geometry Olympiad, 10.5
The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $M$ be the midpoint of arc $\widehat{BAC}$ of the circumcircle, and $P$, $Q$ be the projections of $M$ to the external bisectors of angles $B$ and $C$ respectively. Prove that the line $PQ$ bisects $AD$.
2015 India Regional MathematicaI Olympiad, 8
The length of each side of a convex quadrilateral $ABCD$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.
2022 Bolivia Cono Sur TST, P1
The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.
2022 Kosovo National Mathematical Olympiad, 3
Find all positive integers $n$ such that $10^n+3^n+2$ is a palindrome number.
2016 Harvard-MIT Mathematics Tournament, 26
For positive integers $a,b$, $a\uparrow\uparrow b$ is defined as follows: $a\uparrow\uparrow 1=a$, and $a\uparrow\uparrow b=a^{a\uparrow\uparrow (b-1)}$ if $b>1$.
Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a\uparrow\uparrow 6\not \equiv a\uparrow\uparrow 7$ mod $n$.
2012 China Team Selection Test, 2
Find all integers $k\ge 3$ with the following property: There exist integers $m,n$ such that $1<m<k$, $1<n<k$, $\gcd (m,k)=\gcd (n,k) =1$, $m+n>k$ and $k\mid (m-1)(n-1)$.
2018 Bosnia and Herzegovina Junior BMO TST, 3
Let $\Gamma$ be circumscribed circle of triangle $ABC $ $(AB \neq AC)$. Let $O$ be circumcenter of the triangle $ABC$. Let $M$ be a point where angle bisector of angle $BAC$ intersects $\Gamma$. Let $D$ $(D \neq M)$ be a point where circumscribed circle of the triangle $BOM$ intersects line segment $AM$ and let $E$ $(E \neq M)$ be a point where circumscribed circle of triangle $COM$ intersects line segment $AM$. Prove that $BD+CE=AM$.
1954 Moscow Mathematical Olympiad, 263
Define the maximal value of the ratio of a three-digit number to the sum of its digits.
2012 India IMO Training Camp, 3
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.
1998 Nordic, 4
Let $n$ be a positive integer. Count the number of numbers $k \in \{0, 1, 2, . . . , n\}$ such that $\binom{n}{k}$ is odd. Show that this number is a power of two, i.e. of the form $2^p$ for some nonnegative integer $p$.
1977 Miklós Schweitzer, 9
Suppose that the components of he vector $ \textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $ [a,b]$ with the property that every nontrivial linear combination of them has at most $ n$ zeros in $ [a,b]$. Prove that if $ \sigma$ is an increasing function on $ [a,b]$ and the rank of the operator \[ A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,\] is $ r \leq n$, then $ \sigma$ has exactly $ r$ points of increase.
[i]E. Gesztelyi[/i]
2013 Harvard-MIT Mathematics Tournament, 9
Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
2015 Estonia Team Selection Test, 3
Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$.
a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic.
b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?
1998 Iran MO (3rd Round), 2
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC.
\]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
\]
2012 Saint Petersburg Mathematical Olympiad, 1
Find all integer $b$ such that $[x^2]-2012x+b=0$ has odd number of roots.
2007 AMC 12/AHSME, 2
An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. It is filled with water to a height of $ 40$ cm. A brick with a rectangular base that measures $ 40$ cm by $ 20$ cm and a height of $ 10$ cm is placed in the aquarium. By how many centimeters does the water rise?
$ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$
2009 Canadian Mathematical Olympiad Qualification Repechage, 2
Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$
Cono Sur Shortlist - geometry, 2012.G3
Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.
2006 Mediterranean Mathematics Olympiad, 1
Every point of a plane is colored red or blue, not all with the same color.
Can this be done in such a way that, on every circumference of radius 1,
(a) there is exactly one blue point;
(b) there are exactly two blue points?