Found problems: 85335
2011 AMC 10, 2
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
$ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $
2015 Poland - Second Round, 3
Let $a_{n}=|n(n+1)-19|$ for $n=0, 1, 2, ...$ and $n \neq 4$. Prove that if for every $k<n$ we have $\gcd(a_{n}, a_{k})=1$, then $a_{n}$ is a prime number.
2016 CMIMC, 1
Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle. $F$ and $L$ are two points on the side $[AC]$ such that $AF=LC< AC/2$.
Find the mesure of the angle $\angle FBL$ knowing that $AB^{2}+BC^{2}=AL^{2}+LC^{2}$.
2017 CMIMC Team, 9
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.
2024 USEMO, 5
Let $ABC$ be a scalene triangle whose incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Lines $BE$ and $CF$ meet at $G$. Prove that there exists a point $X$ on the circumcircle of triangle $EFG$ such that the circumcircles of triangles $BCX$ and $EFG$ are tangent, and \[\angle BGC = \angle BXC + \angle EDF.\]
[i]Kornpholkrit Weraarchakul[/i]
2005 Georgia Team Selection Test, 3
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
2023 BMT, 5
Kait rolls a fair $6$-sided die until she rolls a $6$. If she rolls a $6$ on the $N$th roll, she then rolls the die $N$ more times. What is the probability that she rolls a $6$ during these next N times?
2007 JBMO Shortlist, 2
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.
2018 Ecuador Juniors, 3
Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that
$$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$
and determine all cases when equality holds.
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 $x, y\in\mathbb{R}$.