Found problems: 85335
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.
2019 Iran MO (3rd Round), 1
Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that:
$\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $
Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.
1974 IMO Longlists, 12
A circle $K$ with radius $r$, a point $D$ on $K$, and a convex angle with vertex $S$ and rays $a$ and $b$ are given in the plane. Construct a parallelogram $ABCD$ such that $A$ and $B$ lie on $a$ and $b$ respectively, $SA+SB=r$, and $C$ lies on $K$.
2013 National Olympiad First Round, 27
For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$
2014 Rioplatense Mathematical Olympiad, Level 3, 4
A pair (a,b) of positive integers is [i]Rioplatense [/i]if it is true that $b + k$ is a multiple of $a + k$ for all $k \in\{ 0 , 1 , 2 , 3 , 4 \}$. Prove that there is an infinite set $A$ of positive integers such that for any two elements $a$ and $b$ of $A$, with $a < b$, the pair $(a,b)$ is [i]Rioplatense[/i].
2015 Romanian Master of Mathematics, 1
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
2014 Saudi Arabia BMO TST, 4
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )
2022 Purple Comet Problems, 25
Let $ABCD$ be a parallelogram with diagonal $AC = 10$ such that the distance from $A$ to line $CD$ is $6$ and the distance from $A$ to line $BC$ is $7$. There are two non-congruent configurations of $ABCD$ that satisfy these conditions. The sum of the areas of these two parallelograms is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2017-2018 SDPC, 3
Let $n > 2$ be a fixed positive integer. For a set $S$ of $n$ points in the plane, let $P(S)$ be the set of perpendicular bisectors of pairs of distinct points in $S$. Call set $S$ [i]complete[/i] if no two (distinct) pairs of points share the same perpendicular bisector, and every pair of lines in $P(S)$ intersects. Let $f(S)$ be the number of distinct intersection points of pairs of lines in $P(S)$.
(a) Find all complete sets $S$ such that $f(S) = 1$.
(b) Let $S$ be a complete set with $n$ points. Show that if $f(S)>1$, then $f(S) \geq n$.
MOAA Gunga Bowls, 2021.24
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
[i]Proposed by Andy Xu[/i]
2009 Moldova Team Selection Test, 2
[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that
\[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\]
for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]
1994 National High School Mathematics League, 2
Find the 1000th number (from small to large) that is coprime to $105$.
2024 AIME, 1
Every morning, Aya does a $9$ kilometer walk, and then finishes at the coffee shop. One day, she walks at $s$ kilometers per hour, and the walk takes $4$ hours, including $t$ minutes at the coffee shop. Another morning, she walks at $s+2$ kilometers per hour, and the walk takes $2$ hours and $24$ minutes, including $t$ minutes at the coffee shop. This morning, if she walks at $s+\frac12$ kilometers per hour, how many minutes will the walk take, including the $t$ minutes at the coffee shop?
1987 Austrian-Polish Competition, 3
A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.
2019 CHMMC (Fall), 2
Alex, Bob, Charlie, Daniel, and Ethan are five classmates. Some pairs of them are friends. How many possible ways are there for them to be friends such that everyone has at least one friend, and such that there is exactly one loop of friends among the five classmates?
Note: friendship is two-way, so if person x is friends with person y then person y is friends with person $x$.
2001 Singapore Team Selection Test, 3
A game of Jai Alai has eight players and starts with players $P_1$ and $P_2$ on court and the other players $P_3, P_4, P_5, P_6, P_7, P_8$ waiting in a queue. After each point is played, the loser goes to the end of the queue; the winner adds $1$ point to his score and stays on the court; and the player at the head of the queue comes on to contest the next point. Play continues until someone has scored $7$ points. At that moment, we observe that a total of $37$ points have been scored by all eight players. Determine who has won and justify your answer.
2013 Stanford Mathematics Tournament, 3
Karl likes the number $17$ his favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ term has a coefficient of $1$.)
2004 Greece National Olympiad, 1
Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$
$x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$.
1998 Mediterranean Mathematics Olympiad, 1
A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that
\[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]
2014 Bulgaria National Olympiad, 1
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.
[i]Proposed by T. Vitanov, E. Kolev[/i]
2011 F = Ma, 15
A vertical mass-spring oscillator is displaced $\text{2.0 cm}$ from equilibrium. The $\text{100 g}$ mass passes through the equilibrium point with a speed of $\text{0.75 m/s}$. What is the spring constant of the spring?
(A) $\text{90 N/m}$
(B) $\text{100 N/m}$
(C) $\text{110 N/m}$
(D) $\text{140 N/m}$
(E) $\text{160 N/m}$
1991 Arnold's Trivium, 10
Investigate the asymptotic behaviour of the solutions $y$ of the equation $x^5 + x^2y^2 = y^6$ that tend to zero as $x\to0$.