Found problems: 85335
1982 Austrian-Polish Competition, 2
Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.
2006 MOP Homework, 2
Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.
2010 BAMO, 4
Acute triangle $ABC$ has $\angle BAC < 45^\circ$. Point $D$ lies in the interior of triangle $ABC$ so that $BD = CD$ and $\angle BDC = 4 \angle BAC$. Point $E$ is the reflection of $C$ across line $AB$, and point $F$ is the reflection of $B$ across line $AC$. Prove that lines $AD$ and $EF$ are perpendicular.
2021 Kyiv City MO Round 1, 10.1
Prove the following inequality:
$$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$
[i]Proposed by Oleksii Masalitin[/i]
2024 Vietnam National Olympiad, 2
Find all polynomials $P(x), Q(x)$ with real coefficients such that for all real numbers $a$, $P(a)$ is a root of the equation $x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.$
2006 AMC 12/AHSME, 22
Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$?
$ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$
2001 Mongolian Mathematical Olympiad, Problem 2
In an acute-angled triangle $ABC$, $a,b,c$ are sides, $m_a,m_b,m_c$ the corresponding medians, $R$ the circumradius and $r$ the inradius. Prove the inequality
$$\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.$$
2001 AMC 10, 15
A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes.
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$
2006 QEDMO 2nd, 12
Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.
1997 AMC 8, 9
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?
$\textbf{(A)}\ \dfrac{1}{12} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{6} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{2}{3}$
Gheorghe Țițeica 2025, P3
Let $\mathcal{P}_n$ be the set of all real monic polynomial functions of degree $n$. Prove that for any $a<b$, $$\inf_{P\in\mathcal{P}_n}\int_a^b |P(x)|\, dx >0.$$
[i]Cristi Săvescu[/i]
Novosibirsk Oral Geo Oly VIII, 2021.2
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.
2007 District Olympiad, 2
In an urn we have red and blue balls. A person has invented the next game: he extracts balls until he realises for the first time that the number of blue balls is equal to the number of red balls. After a such game he finds out that he has extracted 10 balls, and that there does not exist 3 consecutive balls of the same color. Prove that the fifth and the sixth balls have different collors.
2012 Baltic Way, 16
Let $n$, $m$, and $k$ be positive integers satisfying $(n - 1)n(n + 1) = m^k$. Prove that $k = 1$.
2015 PAMO, Problem 5
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get?
2007 Stanford Mathematics Tournament, 9
Find $a^2+b^2$ given that $a, b$ are real and satisfy \[a=b+\frac{1}{a+\frac{1}{b+\frac{1}{a+\cdots}}}; b=a-\frac{1}{b+\frac{1}{a-\frac{1}{b+\cdots}}}\]
1975 All Soviet Union Mathematical Olympiad, 217
Given a polynomial $P(x)$ with
a) natural coefficients;
b) integer coefficients;
Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.
1962 All-Soviet Union Olympiad, 8
Given is a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $r_1$, to the next closest be $r_2$ and so on, so that the distances from $M$ to the five vertices satisfy $r_1\le r_2\le r_3\le r_4\le r_5$. Find (a) the locus of $M$ which gives $r_3$ the minimum possible value, and (b) the locus of $M$ which gives $r_3$ the maximum possible value.
2016 Taiwan TST Round 1, 1
Let $n$ cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card.
(a)If $n=2015$ , can you make all cards white-side-up through a finite number of moves?
(b)If $n=2016$ , can you make all cards white-side-up through a finite number of moves?
2011 Peru IMO TST, 4
Let $ABC$ be an acute triangle, and $AA_1$, $BB_1$, and $CC_1$ its altitudes. Let $A_2$ be a point on segment $AA_1$ such that $\angle{BA_2C} = 90^{\circ}$. The points $B_2$ and $C_2$ are defined similarly. Let $A_3$ be the intersection point of segments $B_2C$ and $BC_2$. The points $B_3$ and $C_3$ are defined similarly. Prove that the segments $A_2A_3$, $B_2B_3$, and $C_2C_3$ are concurrent.
2009 Today's Calculation Of Integral, 488
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality.
$ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$
2009 APMO, 5
Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90 degrees left turn after every $ \ell$ kilometer driving from start, Rob makes a 90 degrees right turn after every $ r$ kilometer driving from start, where $ \ell$ and $ r$ are relatively prime positive integers.
In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair ($ \ell$, $ r$) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?
2018 NZMOC Camp Selection Problems, 8
Let $\lambda$ be a line and let $M, N$ be two points on $\lambda$. Circles $\alpha$ and $\beta$ centred at $A$ and $B$ respectively are both tangent to $\lambda$ at $M$, with $A$ and $B$ being on opposite sides of $\lambda$. Circles $\gamma$ and $\delta$ centred at $C$ and $D$ respectively are both tangent to $\lambda$ at $N$, with $C$ and $D$ being on opposite sides of $\lambda$. Moreover $A$ and $C$ are on the same side of $\lambda$. Prove that if there exists a circle tangent to all circles $\alpha, \beta, \gamma, \delta$ containing all of them in its interior, then the lines $AC, BD$ and $\lambda$ are either concurrent or parallel.
ABMC Online Contests, 2019 Oct
[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted?
[b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once)
[b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden?
[b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors?
[b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest?
[b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$.
[b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels)
[b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$.
[b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$.
[b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 ELMO Shortlist, G8
Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$.
[i]Andrew Carratu[/i]