Found problems: 85335
MBMT Guts Rounds, 2017
[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide]
[u] Set 1[/u]
[b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$.
[b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$?
[b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for?
[b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$.
[b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school?
[u]Set 2[/u]
[b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make?
[b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have?
[b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$?
[b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$?
[b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point?
[u]Set 3[/u]
[b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD?
[b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s.
[b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$?
[b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits.
[b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$?
PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2015 JBMO Shortlist, 3
Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.
Evangelos Psychas (Greece)
2005 Romania National Olympiad, 2
Let $a,b$ be two integers. Prove that
a) $13 \mid 2a+3b$ if and only if $13 \mid 2b-3a$;
b) If $13 \mid a^2+b^2$ then $13 \mid (2a+3b)(2b+3a)$.
[i]Mircea Fianu[/i]
2019 Poland - Second Round, 1
A cyclic quadrilateral $ABCD$ is given. Point $K_1, K_2$ lie on the segment $AB$, points $L_1, L_2$ on the segment $BC$, points $M_1, M_2$ on the segment $CD$ and points $N_1, N_2$ on the segment $DA$. Moreover, points $K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2$ lie on a circle $\omega$ in that order. Denote by $a, b, c, d$ the lengths of the arcs $N_2K_1, K_2L_1, L_2M_1, M
_2N_1$ of the circle $\omega$ not containing points $K_2, L_2, M_2, N_2$, respectively. Prove that
\begin{align*}
a+c=b+d.
\end{align*}
2008 South africa National Olympiad, 3
Let $a,b,c$ be positive real numbers. Prove that
\[(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)\]
and determine when equality occurs.
2021 AMC 12/AHSME Spring, 1
How many integer values satisfy $|x|<3\pi$?
$\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20$
LMT Guts Rounds, 4
The perimeter of a square is equal in value to its area. Determine the length of one of its sides.
2011 Laurențiu Duican, 2
Let be four real numbers $ x,y,z,t $ satisfying the following system:
$$ \left\{ \begin{matrix} \sin x+\sin y+\sin z +\sin t =0 \\ \cos x+\cos y+\cos z+\cos t=0 \end{matrix} \right. $$
Prove that
$$ \sin ((1+2k)x) +\sin ((1+2k)y) +\sin ((1+2k)z) +\sin ((1+2k)t) =0, $$
for any integer $ k. $
[i]Aurel Bârsan[/i]
2008 IMAC Arhimede, 3
Let $ 0 \leq x \leq 2\pi$. Prove the inequality $ \sqrt {\frac {\sin^{2}x}{1 + \cos^{2}x}} + \sqrt {\frac {\cos^{2}x}{1 + \sin^{2}x}}\geq 1 $
1989 Vietnam National Olympiad, 2
The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.
2010 Malaysia National Olympiad, 8
For any number $x$, let $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. A sequence $a_1,a_2,\cdots$ is given, where \[a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor.\]
How many values of $k$ are there such that $a_k=2010$?
2015 IMAR Test, 2
Let $n$ be a positive integer and let $G_n$ be the set of all simple graphs on $n$ vertices. For each vertex $v$ of a graph in $G_n$, let $k(v)$ be the maximal cardinality of an independent set of neighbours of $v$. Determine $max_{G \in G_n} \Sigma_{v\in V (G)}k(v)$ and the graphs in $G_n$ that achieve this value.
1992 Irish Math Olympiad, 1
Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.
2003 China Team Selection Test, 2
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.
1997 IMO, 4
An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that:
(a) there is no silver matrix for $ n \equal{} 1997$;
(b) silver matrices exist for infinitely many values of $ n$.
2005 Germany Team Selection Test, 2
Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases
[b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$.
[b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$.
In other words, find the Steiner trees of the set $M$ in the above two cases.
2010 Saudi Arabia Pre-TST, 4.2
Let $a$ be a real number.
1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$.
2) Prove that the area of this triangle does not depend on $a$.
1992 Bundeswettbewerb Mathematik, 4
For three sequences $(x_n),(y_n),(z_n)$ with positive starting elements $x_1,y_1,z_1$ we have the following formulae:
\[ x_{n+1} = y_n + \frac{1}{z_n} \quad y_{n+1} = z_n + \frac{1}{x_n} \quad z_{n+1} = x_n + \frac{1}{y_n} \quad (n = 1,2,3, \ldots)\]
a.) Prove that none of the three sequences is bounded from above.
b.) At least one of the numbers $x_{200},y_{200},z_{200}$ is greater than 20.
1956 Putnam, A5
Call a subset of $\{1,2,\ldots, n\}$ [i]unfriendly[/i] if no two of its elements are consecutive. Show that the number of unfriendly subsets with $k$ elements is $\binom{n-k+1}{k}.$
1952 AMC 12/AHSME, 23
If $ \frac {x^2 \minus{} bx}{ax \minus{} c} \equal{} \frac {m \minus{} 1}{m \plus{} 1}$ has roots which are numerical equal but of opposite signs, the value of $ m$ must be:
$ \textbf{(A)}\ \frac {a \minus{} b}{a \plus{} b} \qquad\textbf{(B)}\ \frac {a \plus{} b}{a \minus{} b} \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ \frac {1}{c} \qquad\textbf{(E)}\ 1$
1971 Spain Mathematical Olympiad, 3
If $0 < p$, $0 < q$ and $p +q < 1$ prove $$(px + qy)^2 \le px^2 + qy^2$$
2024 Regional Olympiad of Mexico West, 1
Initially, the numbers $1,3,4$ are written on a board. We do the following process repeatedly. Consider all of the numbers that can be obtained as the sum of $3$ distinct numbers written on the board and that aren't already written, and we write those numbers on the board. We repeat this process, until at a certain step, all of the numbers in that step are greater than $2024$. Determine all of the integers $1\leq k\leq 2024$ that were not written on the board.
2019 Purple Comet Problems, 17
Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.
2016 Regional Olympiad of Mexico Northeast, 6
A positive integer $N$ is called [i]northern[/i] if for each digit $d > 0$, there exists a divisor of $N$ whose last digit is $d$. How many [i]northern [/i] numbers less than $2016$ are there with the fewest number of divisors as possible?
2014 Romania National Olympiad, 1
Find all continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy:
$ \text{(i)}\text{id}+f $ is nondecreasing
$ \text{(ii)} $ There is a natural number $ m $ such that $ \text{id}+f+f^2\cdots +f^m $ is nonincreasing.
Here, $ \text{id} $ represents the identity function, and ^ denotes functional power.