Found problems: 85335
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 $M = abc$
2020 Purple Comet Problems, 26
In $\vartriangle ABC, \angle A = 52^o$ and $\angle B = 57^o$. One circle passes through the points $B, C$, and the incenter of $\vartriangle ABC$, and a second circle passes through the points $A, C$, and the circumcenter of $\vartriangle ABC$. Find the degree measure of the acute angle at which the two circles intersect.
1996 Tournament Of Towns, (520) 3
Let $A', B', C', D', E'$ and $F'$ be the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ of an arbitrary convex hexagon $ABCDEF$ respectively. Express the area of $ABCDEF$ in terms of the areas of the triangles $ABC$, $BCD'$, $CDS'$, $DEF'$, $EFA'$ and $FAB'$.
(A Lopshi tz, NB Vassiliev)
2018-2019 Winter SDPC, 7
In triangle $ABC$, let $D$ be on side $BC$. The line through $D$ parallel to $AB,AC$ meet $AC,AB$ at $E,F$, respectively.
(a) Show that if $D$ varies on line $BC$, the circumcircle of $AEF$ passes through a fixed point $T$.
(b) Show that if $D$ lies on line $AT$, then the circumcircle of $AEF$ is tangent to the circumcircle of $BTC$.
2023 Euler Olympiad, Round 2, 3
Let $ABCD$ be a convex quadrilateral with side lengths satisfying the equality:
$$ AB \cdot CD = AD \cdot BC = AC \cdot BD.$$
Determine the sum of the acute angles of quadrilateral $ABCD$.
[i]Proposed by Zaza Meliqidze, Georgia[/i]
2013 ELMO Shortlist, 8
We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.
For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$.
[i]Proposed by Victor Wang[/i]
1996 Tournament Of Towns, (492) 5
Eight students were asked to solve $8$ problems (the same set of problems for each of the students).
(a) Each problem was solved by $5$ students. Prove that one canfind two students so that each of the problems was solved by at least one of them.
(b) If each problem was solved by $4$ students, then it is possible that no such pair of students exists. Prove this.
(S Tokarev)