Found problems: 85335
2023 Thailand Mathematical Olympiad, 5
Let $\ell$ be a line in the plane and let $90^\circ<\theta<180^\circ$. Consider any distinct points $P,Q,R$ that satisfy the following:
(i) $P$ lies on $\ell$ and $PQ$ is perpendicular to $\ell$
(ii) $R$ lies on the same side of $\ell$ as $Q$, and $R$ doesn’t lie on $\ell$
(iii) for any points $A,B$ on $\ell$, if $\angle ARB=\theta$ then $\angle AQB \geq \theta$.
Find the minimum value of $\angle PQR$.
2012 Estonia Team Selection Test, 1
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
2008 AMC 8, 18
Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?
[asy]
size((150));
draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle);
draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle);
draw((20,0)--(-20,0));
draw((0,20)--(0,-20));
draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow);
draw((-18,1)--(-12, 1), EndArrow);
draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow);
draw((1,-9)--(1,9), EndArrow);
draw((0,12)..(8.3, 8.3)..(12,0), EndArrow);
draw((12,-1)--(18,-1), EndArrow);
label("$A$", (0,20), N);
label("$K$", (20,0), E);
[/asy]
$ \textbf{(A)}\ 10\pi+20\qquad\textbf{(B)}\ 10\pi+30\qquad\textbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \textbf{(E)}\ 20\pi+40$
2010 Moldova National Olympiad, 11.4
Let $ a_n\equal{}1\plus{}\dfrac1{2^2}\plus{}\dfrac1{3^2}\plus{}\cdots\plus{}\dfrac1{n^2}$
Find $ \lim_{n\to\infty}a_n$
2015 AMC 12/AHSME, 10
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$
2011 Northern Summer Camp Of Mathematics, 5
In a meeting, there are $2011$ scientists attending. We know that, every scientist know at least $1509$ other ones. Prove that a group of five scientists can be formed so that each one in this group knows $4$ people in his group.
2010 Junior Balkan Team Selection Tests - Romania, 1
We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.
1992 Tournament Of Towns, (329) 6
A circle is divided into $n$ sectors. Pawns stand on some of the sectors; the total number of pawns equals $n + 1$. This configuration is changed as follows. Any two of the pawns standing on the same sector move simultaneously to the neighbouring sectors in different directions. Prove that after several such transformations a configuration in which no less than half of the sectors are occupied by pawns, will inevitably appear.
(D. Fomin, St Petersburg)
2001 Putnam, 6
Assume that $(a_n)_{n \ge 1}$ is an increasing sequence of positive real numbers such that $\lim \tfrac{a_n}{n}=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\cdots,n-1$?
2005 Junior Tuymaada Olympiad, 8
The sequence of natural numbers is based on the following rule: each term, starting with the second, is obtained from the previous addition works of all its various simple divisors (for example, after the number $12$ should be the number $18$, and after the number $125$ , the number $130$).
Prove that any two sequences constructed in this way have a common member.
1994 All-Russian Olympiad, 5
Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two.
(N. Agakhanov)
2014 MMATHS, Mixer Round
[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have?
[b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$.
[b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular prisms of dimensions $p \times q\times r$ and$ x\times y\times z$ are similar if $\frac{p}{x} = \frac{q}{y} = \frac{r}{z}$ .
[b]p4.[/b] For fixed real values of $p$, $q$, $r$ and $s$, the polynomial $x^4 + px^3 + qx^2 + rx + s$ has four non real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Compute $q$.
[b]p5.[/b] There are $10$ seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat $10$ kids such that no two bad kids are allowed to sit next to each other?
[b]p6.[/b] There are an infinite number of people playing a game. They each pick a different positive integer $k$, and they each win the amount they chose with probability $\frac{1}{k^3}$ . What is the expected amount that all of the people win in total?
[b]p7.[/b] There are $100$ donuts to be split among $4$ teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep?
[b]p8.[/b] Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear:
Dominic: “Sitharthan is not credible.”
Mitchell: “Dominic is not credible.”
Sitharthan: “At least one of Dominic or Mitchell is credible.”
Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!”
They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible.
[b]p9.[/b] Pick an integer $n$ between $1$ and $10$. If no other team picks the same number, we’ll give you $\frac{n}{10}$ points.
[b]p10.[/b] Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you $\frac15$ points for each reasonable argument.
$$(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...$$
[b]p11.[/b] On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Italy TST, 2
Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.
2008 Harvard-MIT Mathematics Tournament, 8
Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.
2007 All-Russian Olympiad, 4
An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer.
[i]A. Golovanov[/i]
1999 Kurschak Competition, 1
For any positive integer $m$, denote by $d_i(m)$ the number of positive divisors of $m$ that are congruent to $i$ modulo $2$. Prove that if $n$ is a positive integer, then
\[\left|\sum_{k=1}^n \left(d_0(k)-d_1(k)\right)\right|\le n.\]
2022 Dutch Mathematical Olympiad, 2
A set consisting of at least two distinct positive integers is called [i]centenary [/i] if its greatest element is $100$. We will consider the average of all numbers in a centenary set, which we will call the average of the set. For example, the average of the centenary set $\{1, 2, 20, 100\}$ is $\frac{123}{4}$ and the average of the centenary set $\{74, 90, 100\}$ is $88$. Determine all integers that can occur as the average of a centenary set.
2016 Saudi Arabia GMO TST, 2
Let $a, b$ be given two real number with $a \ne 0$. Find all polynomials $P$ with real coefficients such that
$x P(x - a) = (x - b)P(x)$ for all $x\in R$
2021 Irish Math Olympiad, 9
Suppose the real numbers $a, A, b, B$ satisfy the inequalities: $$|A - 3a| \le 1 - a\,\,\, , \,\,\, |B -3b| \le 1 - b$$, and $a, b$ are positive. Prove that $$\left|\frac{AB}{3}- 3ab\right | - 3ab \le 1 - ab.$$
2017 Saudi Arabia BMO TST, 3
Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$.
a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent.
b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.
1994 IMO Shortlist, 3
Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?
2018 AIME Problems, 6
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
2006 Hungary-Israel Binational, 3
A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?
2013 Stanford Mathematics Tournament, 7
Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.
2010 Contests, 2
Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$
\[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]