Found problems: 85335
Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$.
[i]Proposed by Michael Kural and Evan Chen[/i]
Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number?
$ \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E)}\ 12 $
$f$ is a one-to-one function from the set of positive integers to itself such that $$f(xy) = f(x) × f(y)$$ Find the minimum possible value of $f(2020)$.
Denote in the triangle $ABC$ by $h$ the length of the height drawn from vertex $A$, and by $\alpha = \angle BAC$. Prove that the inequality $AB + AC \ge BC \cdot \cos \alpha + 2h \cdot \sin \alpha$ . Are there triangles for which this inequality turns into equality?
Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers
\[a_1 + k, a_2 + 2k, \dots, a_p + pk\]
produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$.
[i]Proposed by Ankan Bhattacharya[/i]
Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.
Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.
A rational number $ x$ is called [i]good[/i] if it satisfies: $ x\equal{}\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)\equal{}1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}\minus{}\alpha|\leq\dfrac{1}{2(p\plus{}q)}\] Find all the good numbers.
Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that [list][*] each member in $S$ is a positive integer not exceeding $2002$, [*] if $a,b\in S$ (not necessarily different), then $ab\not\in S$. [/list]
Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\frac{AD}{DB}$ and $\frac{AD}{DC}$. What is $BC$?
Note: The harmonic mean of two numbers $a$ and $b$ is the reciprocal of the arithmetic mean of the reciprocals of $a$ and $b$.
[i]2017 CCA Math Bonanza Lightning Round #4.4[/i]
Let $\omega$ and $O$ be respectively the circumcircle and the circumcenter of a triangle $ABC$. The line $AO$ intersects $\omega$ second time at $A'$. $M_B$ and $M_C$ are the midpoints of $AC$ and $AB$, respectively. The lines $A'M_B$ and $A'M_C$ intersect $\omega$ secondly at points $B'$ and $C$, and also intersect $BC$ at points $D_B$ and $D_C$, respectively. The circumcircles of $CD_BB'$ and $BD_CC'$ intersect at points $P$ and $Q$.
Prove that $O$, $P$, $Q$ are collinear.
[i] (М. Германсков)[/i]
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
Prove:
(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$
(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
Solve in $ \mathbb{N}^2: $
$$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$
Prove that there exist infinitely many positive integers with the following properties:
- it can be written as the sum of $2001$ distinct positive integers,
- it can be written as the sum of $2023$ distinct positive perfect cubes
Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ).
Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.
$ G$ is a group. Prove that the following are equivalent:
1. All subgroups of $ G$ are normal.
2. For all $ a,b\in G$ there is an integer $ m$ such that $ (ab)^m\equal{}ba$.
Let $n$ be the smallest positive integer such that the number obtained by taking $n$’s rightmost digit (decimal expansion) and moving it to be the leftmost digit is $7$ times $n$. Determine the number of digits in $n$.
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?
Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$
[i]Proposed by Stijn Cambie, Belgium [/i]
If $x,y,z$ are positive numbers, prove that
$$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$
Prove: for every positive integer there exists a positive integer having n digits, all of them being $1$'s and $2$'s only, such that this number is divisible by $2^{n}$.
Is this still true in base $4$ or $6$¿
The number devil has coloured the integer numbers: every integer is coloured either black or white.
The number $1$ is coloured white. For every two white numbers $a$ and $b$ ($a$ and $b$ are allowed to be equal) the numbers $a-b$ and $a + $b have di fferent colours.
Prove that $2011$ is coloured white.