Found problems: 85335
2006 Pre-Preparation Course Examination, 5
Powers of $2$ in base $10$ start with $3$ or $4$ more frequently? What is their state in base $3$? First write down an exact form of the question.
2023 Taiwan TST Round 2, C
Integers $n$ and $k$ satisfy $n > 2023k^3$. Kingdom Kitty has $n$ cities, with at most one road between each pair of cities. It is known that the total number of roads in the kingdom is at least $2n^{3/2}$. Prove that we can choose $3k + 1$ cities such that the total number of roads with both ends being a chosen city is at least $4k$.
2014 PUMaC Combinatorics A, 8
There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends’ homes.
1981 AMC 12/AHSME, 10
The lines $L$ and $K$ are symmetric to each other with respect to the line $y=x$. If the equation of the line $L$ is $y=ax+b$ with $a\neq 0$ and $b \neq 0$, then the equation of $K$ is $y=$
$\text{(A)}\ \frac 1ax+b \qquad \text{(B)}\ -\frac 1ax+b \qquad \text{(C)}\ \frac 1ax - \frac ba \qquad \text{(D)}\ \frac 1ax+\frac ba \qquad \text{(E)}\ \frac 1ax -\frac ba$
1995 Portugal MO, 4
The diameter $[AC]$ of a circle is divided into four equal segments by points $P, M$ and $Q$. Consider a segment $[BD]$ that passes through $P$ and cuts the circle at $B$ and $D$, such that $PD =\frac{3}{2} AP$. Knowing that the area of the triangle $[ABP]$ has measure $1$ cm$^2$ , calculate the area of $[ABCD]$?
[img]https://1.bp.blogspot.com/-ibre0taeRo8/X4KiWWSROEI/AAAAAAAAMl4/xFNfpQBxmMMVLngp5OWOXRLMuaxf3nolQCLcBGAsYHQ/s154/1995%2Bportugal%2Bp5.png[/img]
PEN E Problems, 29
Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.
2017 Romania Team Selection Test, P4
Determine the smallest radius a circle passing through EXACTLY three lattice points may have.
1960 AMC 12/AHSME, 12
The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
$ \textbf{(A) }\text{a point}\qquad\textbf{(B) }\text{ a straight line} \qquad\textbf{(C) }\text{two straight lines}\qquad\textbf{(D) }\text{a circle}\qquad$
$\textbf{(E) }\text{two circles} $
PEN P Problems, 40
Show that [list=a][*] infinitely many perfect squares are a sum of a perfect square and a prime number, [*] infinitely many perfect squares are not a sum of a perfect square and a prime number. [/list]
2011 AMC 10, 2
Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
$ \textbf{(A)}\ 80 \qquad
\textbf{(B)}\ 82 \qquad
\textbf{(C)}\ 85 \qquad
\textbf{(D)}\ 90 \qquad
\textbf{(E)}\ 95 $
2008 Iran MO (3rd Round), 1
Suppose that $ f(x)\in\mathbb Z[x]$ be an irreducible polynomial. It is known that $ f$ has a root of norm larger than $ \frac32$. Prove that if $ \alpha$ is a root of $ f$ then $ f(\alpha^3\plus{}1)\neq0$.
2007 Harvard-MIT Mathematics Tournament, 5
Five marbles of various sizes are placed in a conical funnel. Each marble is in contact with the adjacent marble(s). Also, each marble is in contact all around the funnel wall. The smallest marble has a radius of $8$, and the largest marble has a radius of $18$. What is the radius of the middle marble?
2019 IMEO, 1
Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$.
[i]Proposed by Alexandru Lopotenco (Moldova)[/i]
2004 China Girls Math Olympiad, 4
A deck of $ 32$ cards has $ 2$ different jokers each of which is numbered $ 0$. There are $ 10$ red cards numbered $ 1$ through $ 10$ and similarly for blue and green cards. One chooses a number of cards from the deck. If a card in hand is numbered $ k$, then the value of the card is $ 2^k$, and the value of the hand is sum of the values of the cards in hand. Determine the number of hands having the value $ 2004$.
2011 Indonesia TST, 1
Find all real number $x$ which could be represented as
$x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$
2023 VN Math Olympiad For High School Students, Problem 9
Given a quadrilateral $ABCD$ inscribed in $(O)$. Let $L, J$ be the [i]Lemoine[/i] point of $\triangle ABC$ and $\triangle ACD$.
Prove that: $AC, BD, LJ$ are concurrent.
1956 Moscow Mathematical Olympiad, 322
A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.
2004 District Olympiad, 1
From a fixed set formed by the first consecutive natural numbers, find the number of subsets having exactly three elements, and these in arithmetic progression.
2015 Romania Team Selection Tests, 3
Let $n$ be a positive integer . If $\sigma$ is a permutation of the first $n$ positive integers , let $S(\sigma)$ be the set of all distinct sums of the form $\sum_{i=k}^{l} \sigma(i)$ where $1 \leq k \leq l \leq n$ .
[b](a)[/b] Exhibit a permutation $\sigma$ of the first $n$ positive integers such that $|S(\sigma)|\geq \left \lfloor{\frac{(n+1)^2}{4}}\right \rfloor $.
[b](b)[/b] Show that $|S(\sigma)|>\frac{n\sqrt{n}}{4\sqrt{2}}$ for all permutations $\sigma$ of the first $n$ positive integers .
2016 India PRMO, 14
Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.
2001 Baltic Way, 12
Let $a_1, a_2,\ldots , a_n$ be positive real numbers such that $\sum_{i=1}^na_i^3=3$ and $\sum_{i=1}^na_i^5=5$. Prove that $\sum_{i=1}^na_i>\frac{3}{2}$.
LMT Speed Rounds, 9
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
[i]Proposed byMuztaba Syed[/i]
2014 Online Math Open Problems, 7
Define the function $f(x, y, z)$ by\[f(x, y, z) = x^{y^z} - x^{z^y} + y^{z^x} - y^{x^z} + z^{x^y}.\]Evaluate $f(1, 2, 3) + f(1, 3, 2) + f(2, 1, 3) + f(2, 3, 1) + f(3, 1, 2) + f(3, 2, 1)$.
[i]Proposed by Robin Park[/i]
2010 China Team Selection Test, 2
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
2011 Hanoi Open Mathematics Competitions, 5
Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$.
Find the greatest value of $M = abc$