Found problems: 85335
1974 Bundeswettbewerb Mathematik, 3
Let $M$ be a set with $n$ elements. How many pairs $(A, B)$ of subsets of $M$ are there such that $A$ is a subset of $B?$
1998 Iran MO (2nd round), 1
If $a_1<a_2<\cdots<a_n$ be real numbers, prove that:
\[ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. \]
1964 All Russian Mathematical Olympiad, 052
Given an expression $$x_1 : x_2 : ... : x_n$$ ( $:$ means division). We can put the braces as we want. How many expressions can we obtain?
1945 Moscow Mathematical Olympiad, 095
Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.
2020 CCA Math Bonanza, L5.4
Submit a positive integer less than or equal to $15$. Your goal is to submit a number that is close to the number of teams submitting it. If you submit $N$ and the total number of teams at the competition (including your own team) who submit $N$ is $T$, your score will be $\frac{2}{0.5|N-T|+1}$.
[i]2020 CCA Math Bonanza Lightning Round #5.4[/i]
DMM Team Rounds, 2003
[b]p1.[/b] In a $3$-person race, how many different results are possible if ties are allowed?
[b]p2.[/b] An isosceles trapezoid has lengths $5$, $5$, $5$, and $8$. What is the sum of the lengths of its diagonals?
[b]p3.[/b] Let $f(x) = (1 + x + x^2)(1 + x^3 + x^6)(1 + x^9 + x^{18})...$. Compute $f(4/5)$.
[b]p4.[/b] Compute the largest prime factor of $3^{12} - 1$.
[b]p5.[/b] Taren wants to throw a frisbee to David, who starts running perpendicular to the initial line between them at rate $1$ m/s. Taren throws the frisbee at rate $2$ m/s at the same instant David begins to run. At what angle should Taren throw the frisbee?
[b]p6.[/b] The polynomial $p(x)$ leaves remainder $6$ when divided by $x-5$, and $5$ when divided by $x-6$. What is the remainder when $p(x)$ is divided by $(x - 5)(x - 6)$?
[b]p7.[/b] Find the sum of the cubes of the roots of $x^{10} + x^9 + ... + x + 1 = 0$.
[b]p8.[/b] A circle of radius $1$ is inscribed in a the parabola $y = x^2$. What is the $x$-coordinate of the intersection in the first quadrant?
[b]p9.[/b] You are stuck in a cave with $3$ tunnels. The first tunnel leads you back to your starting point in $5$ hours, and the second tunnel leads you back there in $7$ hours. The third tunnel leads you out of the cave in $4$ hours. What is the expected number of hours for you to exit the cave, assuming you choose a tunnel randomly each time you come across your point of origin?
[b]p10.[/b] What is the minimum distance between the line $y = 4x/7 + 1/5$ and any lattice point in the plane? (lattice points are points with integer coordinates)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 Azerbaijan BMO TST, 4
Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.
2023 Thailand Online MO, 4
Let $ABC$ be a triangle, and let $D$ and $D_1$ be points on segment $BC$ such that $BD = CD_1$. Construct point $E$ such that $EC\perp BC$ and $ED\perp AC$. Similarly, construct point $F$ such that $FB\perp BC$ and $FD\perp AB$. Prove that $EF\perp AD_1$.
2024 HMNT, 16
Compute $$\frac{2+3+\cdots+100}{1}+\frac{3+4+\cdots+100}{1+2}+\cdots+\frac{100}{1+2+\cdots+99}.$$
2010 Contests, 2
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.
2003 China Team Selection Test, 2
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.
2013 Putnam, 2
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.
2009 Romania National Olympiad, 1
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and
$$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$
1991 Arnold's Trivium, 78
Solve the Cauchy problem
\[\frac{\partial ^2A}{\partial t^2}=9\frac{\partial^2 A}{\partial x^2}-2B,\;\frac{\partial^2 B}{\partial t^2}=6\frac{\partial^2 B}{\partial x^2}-2A\]
\[A|_{t=0}=\cos x,\; B|_{t=0}=0,\; \left.\frac{\partial A}{\partial t}\right|_{t=0}=\left.\frac{\partial B}{\partial t}\right|_{t=0}=0\]
2007 Princeton University Math Competition, 2
Find the biggest non-integer $x$ such that $(x+2)^2 + (x+3)^3 + (x+4)^4 = 2$.
2011 Tournament of Towns, 6
Two ants crawl along the sides of the $49$ squares of a $7 * 7$ board. Each ant passes through
all $64$ vertices exactly once and returns to its starting point. What is the smallest possible
number of sides covered by both ants?
1989 Tournament Of Towns, (226) 4
Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$
(G. Galperin)
2020 Online Math Open Problems, 5
Compute the smallest positive integer $n$ such that there do not exist integers $x$ and $y$ satisfying $n=x^3+3y^3$.
[i]Proposed by Luke Robitaille[/i]
2014 ASDAN Math Tournament, 2
Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers.
1994 Poland - First Round, 3
A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.
1987 Tournament Of Towns, (139) 4
Angle $A$ of the acute-angled triangle $ABC$ equals $60^o$ . Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$, passes through the circumcircle 's centre.
(V . Pogrebnyak , year 12 student , Vinnitsa,)
2019 Costa Rica - Final Round, 4
Let $g: R \to R$ be a linear function such that $g (1) = 0$. If $f: R \to R$ is a quadratic function such what $g (x^2) = f (x)$ and $f (x + 1) - f (x - 1) = x$ for all $x \in R$. Determine the value of $f (2019)$.
2014 Math Hour Olympiad, 8-10.2
A complete set of the Encyclopedia of Mathematics has $10$ volumes. There are ten mathematicians in Mathemagic Land, and each of them owns two volumes of the Encyclopedia. Together they own two complete sets. Show that there is a way for each mathematician to donate one book to the library such that the library receives a complete set.
2011 Swedish Mathematical Competition, 2
Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$
1983 Spain Mathematical Olympiad, 6
In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost $1$ shilling and $2$ pence, and a glass of lemonade, four sandwiches and $10$ biscuits they are worth $1$ shilling and $5$ pence. Find the price of:
a) a glass of lemonade, a sandwich and a cake;
b) two glasses of lemonade, three sandwiches and five biscuits.
($1$ shilling = $12$ pence).