Found problems: 85335
2020 USA IMO Team Selection Test, 1
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?
[i]Carl Schildkraut and Milan Haiman[/i]
2023 Girls in Mathematics Tournament, 1
Define $(a_n)$ a sequence, where $a_1= 12, a_2= 24$ and for $n\geq 3$, we have: $$a_n= a_{n-2}+14$$
a) Is $2023$ in the sequence?
b) Show that there are no perfect squares in the sequence.
2014 AMC 10, 11
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$
Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$
Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$
For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?
$\textbf{(A) }\$179.95\qquad
\textbf{(B) }\$199.95\qquad
\textbf{(C) }\$219.95\qquad
\textbf{(D) }\$239.95\qquad
\textbf{(E) }\$259.95\qquad$
2015 BMT Spring, 18
A value $x \in [0, 1]$ is selected uniformly at random. A point $(a, b)$ is called [i]friendly [/i] to $x$ if there exists a circle between the lines $y = 0$ and $y = 1$ that contains both $(a, b)$ and $(0, x)$. Find the area of the region of the plane determined by possible locations of friendly points.
2006 Tournament of Towns, 4
Anna, Ben and Chris sit at the round table passing and eating nuts. At first only Anna has the nuts that she divides equally between Ben and Chris, eating a leftover (if there is any). Then Ben does the same with his pile. Then Chris does the same with his pile. The process repeats itself: each of the children divides his/her pile of nuts equally between his/her neighbours eating the leftovers if there are any. Initially, the number of nuts is large enough (more than 3). Prove that
a) at least one nut is eaten; [i](3 points)[/i]
b) all nuts cannot be eaten. [i](3 points)[/i]
2019 Thailand Mathematical Olympiad, 9
A [i]chaisri[/i] figure is a triangle which the three vertices are vertices of a regular $2019$-gon. Two different chaisri figure may be formed by different regular $2019$-gon.
A [i]thubkaew[/i] figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.
Determine the maximum number of vertices that a thubkaew figure may have.
1991 China National Olympiad, 2
Given $I=[0,1]$ and $G=\{(x,y)|x,y \in I\}$, find all functions $f:G\rightarrow I$, such that $\forall x,y,z \in I$ we have:
i. $f(f(x,y),z)=f(x,f(y,z))$;
ii. $f(x,1)=x, f(1,y)=y$;
iii. $f(zx,zy)=z^kf(x,y)$.
($k$ is a positive real number irrelevant to $x,y,z$.)
1978 Vietnam National Olympiad, 3
The triangle $ABC$ has angle $A = 30^o$ and $AB = \frac{3}{4} AC$. Find the point $P$ inside the triangle which minimizes $5 PA + 4 PB + 3 PC$.
2019 Baltic Way, 5
The $2m$ numbers
$$1\cdot 2, 2\cdot 3, 3\cdot 4,\hdots,2m(2m+1)$$
are written on a blackboard, where $m\geq 2$ is an integer. A [i]move[/i] consists of choosing three numbers $a, b, c$, erasing them from the board and writing the single number
$$\frac{abc}{ab+bc+ca}$$
After $m-1$ such moves, only two numbers will remain on the blackboard. Supposing one of these is $\tfrac{4}{3}$, show that the other is larger than $4$.
1975 USAMO, 4
Two given circles intersect in two points $ P$ and $ Q$. Show how to construct a segment $ AB$ passing through $ P$ and terminating on the circles such that $ AP \cdot PB$ is a maximum.
1979 IMO Longlists, 55
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.
1996 Tournament Of Towns, (484) 2
Does there exist an integer n such that all three numbers
(a) $n - 96$, $n$ and $n + 96$
(b) $n - 1996$, $n$ and $n + 1996$
are positive prime numbers?
(V Senderov)
2011 Mongolia Team Selection Test, 3
Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa)
(proposed by B. Batbaysgalan, folklore).
2023 China Team Selection Test, P22
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$
2025 Bulgarian Winter Tournament, 12.1
Let $a,b,c$ be positive real numbers with $a+b>c$. Prove that $ax + \sin(bx) + \cos(cx) > 1$ for all $x\in \left(0, \frac{\pi}{a+b+c}\right)$.
2024 Korea - Final Round, P6
Prove that there exists a positive integer $K$ that satisfies the following condition.
Condition: For any prime $p > K$, the number of positive integers $a \le p$ that $p^2 \mid a^{p-1} - 1$ is less than $\frac{p}{2^{2024}}$
1997 Italy TST, 4
There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?
2018 HMIC, 5
Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.
2002 Junior Balkan Team Selection Tests - Romania, 1
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.
2004 Greece JBMO TST, 4
Let $a,b$ be positive real numbers such that $b^3+b\le a-a^3$. Prove that:
i) $b<a<1$
ii) $a^2+b^2<1$
2013 Indonesia MO, 5
Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that:
- $P(x) = P_1(x) + P_2(x) + P_3(x)$
- $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root)
- The roots of $P_1, P_2, P_3$ are different
2024 AMC 12/AHSME, 12
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$
2020 Brazil Cono Sur TST, 2
A number n is called charming when $ 4k^2 + n $ is a prime number for every $ 0 \leq k <n $ integer, find all charming numbers.
2022 Caucasus Mathematical Olympiad, 6
16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it happens that $k$ teams have non-negative balance of wins (for example, the team that won in the first round with a score of 4-2 and lost in the second with a score of 4-3 fits the condition: it has $4+3=7$ wins and $2+4=6$ losses). Find the least possible $k$.
2009 Bundeswettbewerb Mathematik, 2
Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent:
(A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$
(B) The number $n$ has at least two different prime divisors.