Found problems: 85335
Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.
Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$
Find the 1000th term of the sequence.
Prove: If the sum of all positive divisors of $ n \in \mathbb{Z}^{\plus{}}$ is a power of two, then the number/amount of the divisors is a power of two.
In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to $a$ and $b$.
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions:
(i) For any positive integer $n$, $b_n \leq c_n$;
(ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.
Roma wrote on the board each of the numbers $2018, 2019, 2020$, $100$ times each. Let us denote by $S(n)$ the sum of digits of positive integer $n$. In one action, Roma can choose any positive integer $k$ and instead of any three numbers $a, b, c$ written on the board write the numbers $2S(a + b) + k, 2S(b + c) + k$ and $2S(c + a) + k$. Can Roma after several such actions make $299$ numbers on the board equal, and the last one differing from them by $1$?
[i]Proposed by Oleksii Masalitin[/i]
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
Given a $\triangle ABC$ with circumcenter $O$ and orthocenter $H(O\ne H)$. Denote the midpoints of $BC, AC$ as $D, E$ and let $D', E'$ be the reflections of $D, E$ w.r.t. point $H$, respectively. If lines $AD'$ and $BE'$ meet at $K$, compute $\frac{KO}{KH}$.
Francesca and Giorgia play the following game. On a table there are initially coins piled up in some stacks, possibly in different numbers in each stack, but with at least one coin. In turn, each player chooses exactly one move between the following:
(i) she chooses a stack that has an even non-zero number of coins $ 2k$ and breaks it into two identical stacks of coins, i.e. each stack contains $ k$ coins;
(ii) she removes from the table the stacks with coins in an odd number, i.e. all such in odd number, not just those with a specific odd number.
At each turn, a player necessarily moves: if one choice is not available, the she must take the other.
Francesca moves first. The one who removes the last coin from the table wins.
1. If initially there is only one stack of coins on the table, and this stack contains $ 2008^{2008}$ coins, which of the players has a winning strategy?
2. For which initial configurations of stacks of coins does Francesca have a winning strategy?
What's the greatest integer $n$ for which the system $k < x^k < k + 1$ for $k = 1,2,..., n$ has a solution?
Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.
Show that for any positive real numbers $a$ and $b$ the following inequality hold,
$$\frac{a(a+1)}{b+1}+\frac{b(b+1)}{a+1}\geq a+b.$$
Consider the "tower of power" $2^{2^{2^{.^{.^{.^2}}}}}$, where there are $2007$ twos including the base. What is the last (units) digit of this number?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$
$\textbf{(J) }9\hspace{14.3em}\textbf{(K) }2007$
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the [i]product[/i] of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are $8^6=262144$ participants, and no two get the same score on every problem. Find the score of the participant whose rank was $7^6 = 117649$.
[i]Proposed by Yang Liu[/i]
John, Paul, George, and Ringo baked a circular pie. Each cut a piece that was a sector of the circle. John took one-third of the whole pie. Paul took one-fourth of the whole pie. George took one-fifth of the whole pie. Ringo took one-sixth of the whole pie. At the end the pie had one sector remaining. Find the measure in degrees of the angle formed by this remaining sector.
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called [i]lucky[/i], if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value).
(a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$
(b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$
Proposed by Ilya Bogdanov
For a sequence, one can perform the following operation: select three adjacent terms $a,b,c,$ and change it into $b,c,a.$ Determine all the possible positive integers $n\geq 3,$ such that after finite number of operation, the sequence $1,2,\cdots, n$ can be changed into $n,n-1,\cdots,1$ finally.
Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
[i]Proposed by Igor Voronovich, Belarus[/i]
Show that if real numbers $x<1<y$ satisfy the inequality
$$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.
Find the number of integers $n$ with $1\le n\le 100$ for which $n-\phi(n)$ is prime. Here $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$.
[i]Proposed by Mehtaab Sawhney[/i]
The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$