This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2010 LMT, 15
Tags:
Determine the number of ordered pairs $(x,y)$ with $x$ and $y$ integers between $-5$ and $5,$ inclusive, such that $(x+y)(x+3y)=(x+2y)^2.$

2017 NIMO Summer Contest, 8
Tags:
Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? (Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.) [i]Proposed by Michael Tang[/i]

1995 IMC, 3
Tags: real analysis , differentiable function
Let $f$ be twice continuously differentiable on $(0,\infty)$ such that $\lim_{x \to 0^{+}}f'(x)=-\infty$ and $\lim_{x \to 0^{+}}f''(x)=\infty$. Show that $$\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.$$

1953 AMC 12/AHSME, 10
Tags:
The number of revolutions of a wheel, with fixed center and with an outside diameter of $ 6$ feet, required to cause a point on the rim to go one mile is: $ \textbf{(A)}\ 880 \qquad\textbf{(B)}\ \frac{440}{\pi} \qquad\textbf{(C)}\ \frac{880}{\pi} \qquad\textbf{(D)}\ 440\pi \qquad\textbf{(E)}\ \text{none of these}$

2016 Estonia Team Selection Test, 11
Tags: number theory , perfect square
Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

2014 Czech-Polish-Slovak Junior Match, 3
Tags: combinatorics , tiles , tiling
We have $10$ identical tiles as shown. The tiles can be rotated, but not flipper over. A $7 \times 7$ board should be covered with these tiles so that exactly one unit square is covered by two tiles and all other fields by one tile. Designate all unit sqaures that can be covered with two tiles. [img]https://cdn.artofproblemsolving.com/attachments/d/5/6602a5c9e99126bd656f997dee3657348d98b5.png[/img]

2014 Puerto Rico Team Selection Test, 5
Tags: combinatorics
In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)

2000 Kazakhstan National Olympiad, 4
Tags: number theory , diophantine equation , diophantine
Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $

2018 Bosnia and Herzegovina Junior BMO TST, 2
Tags: number theory , diophantine equation , exponential equation , prime number
Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

VI Soros Olympiad 1999 - 2000 (Russia), 11.4
Tags: geometry , 3d geometry , sphere
Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.

1976 Kurschak Competition, 2
Tags: combinatorics
A lottery ticket is a choice of $5$ distinct numbers from $1, 2,3,...,90$. Suppose that $5^5$ distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the four.

2002 Singapore Team Selection Test, 2
Tags: number theory , floor function , integer , divide , divisible
For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

2013 Iran MO (2nd Round), 1
Tags: number theory
Find all pairs $(a,b)$ of positive integers for which $\gcd(a,b)=1$, and $\frac{a}{b}=\overline{b.a}$. (For example, if $a=92$ and $b=13$, then $b/a=13.92$ )

2011 Mathcenter Contest + Longlist, 7 sl9
Tags: functional equation , algebra , functional
Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$ [i](-InnoXenT-)[/i]

2020 HK IMO Preliminary Selection Contest, 4
Tags: algebra , digit
In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be chosen in order to maximise the score?

Kvant 2024, M2801
Tags: game , combinatorics
Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

1991 Arnold's Trivium, 87
Tags: derivative , function , calculus , quadratic , vector , linear algebra , matrix
Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

2009 JBMO Shortlist, 1
Tags: number theory
Solve in non-negative integers the equation $ 2^{a}3^{b} \plus{} 9 \equal{} c^{2}$

2017 BMT Spring, 8
Tags: combinatorics
In a $1024$ person randomly seeded single elimination tournament bracket, each player has a unique skill rating. In any given match, the player with the higher rating has a $\frac34$ chance of winning the match. What is the probability the second lowest rated player wins the tournament?

2013 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , divide , divisible
Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns

2002 National Olympiad First Round, 12
Tags:
What is the least possible value of $ab + bc + ac$ such that $a^2 + b^2 + c^2 = 1$ where $a,b,c$ are real numbers? $ \textbf{a)}\ -1 \qquad\textbf{b)}\ -\dfrac 12 \qquad\textbf{c)}\ -\dfrac 13 \qquad\textbf{d)}\ -\dfrac{1}{2\sqrt 2} \qquad\textbf{e)}\ 0 $

2003 Gheorghe Vranceanu, 1
Tags: matrix , linear algebra , cayley-hamilton
Prove that if a $ 2\times 2 $ complex matrix has the property that there exists a natural number $ n $ such that $ \text{tr}\left( A^n\right) =\text{tr}\left( A^{n+1} \right) =0, $ then $ A^2=0. $

2010 Kosovo National Mathematical Olympiad, 2
Tags: algebra
The set $S\subseteq \mathbb{R}$ is given with the properties: $(a) \mathbb{Z}\subset S$, $(b) (\sqrt 2 +\sqrt 3)\in S$, $(c)$ If $x,y\in S$ then $x+y\in S$, and $(d)$ If $x,y\in S$ then $x\cdot y\in S$. Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.

2022 HMNT, 2
Tags: geometry , combinatorics
What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?

2010 IMO Shortlist, 1
Tags: combinatorics , poset , linear extensions , counting
In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? [i]Proposed by Gerhard Wöginger, Austria[/i]