Found problems: 85335
2007 Moldova Team Selection Test, 1
Let $a_{1}, a_{2}, \ldots, a_{n}\in [0;1]$. If $S=a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}$ then prove that \[\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+\ldots+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}\]
2004 AMC 12/AHSME, 19
Circles $ A$, $ B$ and $ C$ are externally tangent to each other and internally tangent to circle $ D$. Circles $ B$ and $ C$ are congruent. Circle $ A$ has radius $ 1$ and passes through the center of $ D$. What is the radius of circle $ B$?
[asy]
unitsize(15mm);
pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
draw(Circle(D,2));
draw(Circle(A,1));
draw(Circle(B,8/9));
draw(Circle(C,8/9));
label("\(A\)", A);
label("\(B\)", B);
label("\(C\)", C);
label("D", (-1.2,1.8));[/asy]
$ \textbf{(A)}\ \frac23 \qquad \textbf{(B)}\ \frac {\sqrt3}{2} \qquad \textbf{(C)}\ \frac78 \qquad \textbf{(D)}\ \frac89 \qquad \textbf{(E)}\ \frac {1 \plus{} \sqrt3}{3}$
2019 Switzerland Team Selection Test, 10
Let $n \geq 5$ be an integer. A shop sells balls in $n$ different colors. Each of $n + 1 $ children bought three balls with different colors, but no two children bought exactly the same color combination. Show that there are at least two children who bought exactly one ball of the same color.
2006 Iran Team Selection Test, 3
Let $l,m$ be two parallel lines in the plane.
Let $P$ be a fixed point between them.
Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$.
(By angle $EPF$ we mean the directed angle)
Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.
2019 CCA Math Bonanza, I5
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
[i]2019 CCA Math Bonanza Individual Round #5[/i]
2003 Tournament Of Towns, 4
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
2015 JBMO TST - Turkey, 5
A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.
2012 Singapore Junior Math Olympiad, 5
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
2013 Brazil Team Selection Test, 2
Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.
2016 Israel Team Selection Test, 1
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
2014 Purple Comet Problems, 18
Find the number of subsets of $\{1,3,5,7,9,11,13,15,17,19\}$ where the elements in the subset add to $49$.
2003 VJIMC, Problem 1
Let $d(k)$ denote the number of natural divisors of a natural number $k$. Prove that for any natural number $n_0$ the sequence $\left\{d(n^2+1)\right\}^\infty_{n=n_0}$ is not strictly monotone.
2006 Australia National Olympiad, 2
Let $f$ be a function defined on the positive integers, taking positive integral values, such that
$f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$,
$f(a) < f(b)$ if $a < b$,
$f(3) \geq 7$.
Find the smallest possible value of $f(3)$.
1963 AMC 12/AHSME, 21
The expression $x^2-y^2-z^2+2yz+x+y-z$ has:
$\textbf{(A)}\ \text{no linear factor with integer coeficients and integer exponents} \qquad$
$
\textbf{(B)}\ \text{the factor }-x+y+z \qquad$
$
\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad$
$
\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad$
$
\textbf{(E)}\ \text{the factor }x-y+z+1$
2015 Princeton University Math Competition, A5
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that the sum
\[a+a^2+a^3+\cdots+a^{(p-2)!} \]is not divisible by $p$?
2007 Indonesia TST, 2
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
\[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\]
for all real numbers $x$ and $y$.
1959 AMC 12/AHSME, 3
If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:
$ \textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these} $
IV Soros Olympiad 1997 - 98 (Russia), 11.5
Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.
2021 Federal Competition For Advanced Students, P2, 3
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied.
(Walther Janous)
1991 Bulgaria National Olympiad, Problem 3
Prove that for every prime number $p\ge5$,
(a) $p^3$ divides $\binom{2p}p-2$;
(b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.
2023 Regional Competition For Advanced Students, 4
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
1976 AMC 12/AHSME, 29
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is
$\textbf{(A) }22\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }28$
1973 AMC 12/AHSME, 18
If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder
$ \textbf{(A)}\ \text{never} \qquad
\textbf{(B)}\ \text{sometimes only} \qquad
\textbf{(C)}\ \text{always} \qquad$
$ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad
\textbf{(E)}\ \text{none of these}$
2014 Albania Round 2, 2
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
2006 India Regional Mathematical Olympiad, 7
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$