Found problems: 85335
2004 Romania National Olympiad, 1
Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$.
[i]Lucian Dragomir[/i]
V Soros Olympiad 1998 - 99 (Russia), 11.2
Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality
$$(x + y +z + u)^2 \ge Cyz .$$
1991 AMC 8, 1
$1,000,000,000,000-777,777,777,777=$
$\text{(A)}\ 222,222,222,222 \qquad \text{(B)}\ 222,222,222,223 \qquad \text{(C)}\ 233,333,333,333 \\ \text{(D)}\ 322,222,222,223 \qquad \text{(E)}\ 333,333,333,333$
1973 AMC 12/AHSME, 27
Cars A and B travel the same distance. Care A travels half that [i]distance[/i] at $ u$ miles per hour and half at $ v$ miles per hour. Car B travels half the [i]time[/i] at $ u$ miles per hour and half at $ v$ miles per hour. The average speed of Car A is $ x$ miles per hour and that of Car B is $ y$ miles per hour. Then we always have
$ \textbf{(A)}\ x \leq y\qquad
\textbf{(B)}\ x \geq y \qquad
\textbf{(C)}\ x\equal{}y \qquad
\textbf{(D)}\ x<y\qquad
\textbf{(E)}\ x>y$
2021 LMT Fall, 10
There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.
2011 Bundeswettbewerb Mathematik, 2
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.
2008 Korean National Olympiad, 1
Let $V=[(x,y,z)|0\le x,y,z\le 2008]$ be a set of points in a 3-D space.
If the distance between two points is either $1, \sqrt{2}, 2$, we color the two points differently.
How many colors are needed to color all points in $V$?
2021 DIME, 1
Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$.
[i]Proposed by pog[/i]
2014 CHMMC (Fall), 2
Consider two overlapping regular tetrahedrons of side length $2$ in space. They are centered at the same point, and the second one is oriented so that the lines from its center to its vertices are perpendicular to the faces of the first tetrahedron. What is the volume encompassed by the combined solid?
1967 Polish MO Finals, 1
Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.
2004 Iran Team Selection Test, 1
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that:
\[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]
2022 Bundeswettbewerb Mathematik, 4
For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that:
(a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$.
(b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.
2020-21 IOQM India, 17
How many two digit numbers have exactly $4$ positive factors? $($Here $1$ and the number $n$ are also considered as factors of $n. )$
2014 Swedish Mathematical Competition, 5
In next year's finals in Schools Mathematics competition, $20$ finalists will participate. The final exam contains six problems. Emil claims that regardless of results, there must be five contestants and two problems such that either all the five contestants solve both problems, or neither of them solve any of the two problems. Is he right?
2019 HMNT, 3
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
2003 Tournament Of Towns, 4
In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.
2013 NIMO Summer Contest, 8
A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i]
2024 LMT Fall, 24
Find the number of positive integers $x$ that satisfy
\[ \left \lfloor{\frac{2024}{ \left \lfloor \frac{2024}{x} \right \rfloor }} \right \rfloor = x.\]
1966 IMO Shortlist, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
2009 USAMTS Problems, 3
Prove that if $a$ and $b$ are positive integers such that $a^2 + b^2$ is a multiple of $7^{2009}$, then $ab$ is a multiple of $7^{2010}$.
1987 Greece National Olympiad, 3
There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$
2019 CMIMC, 14
Consider the following function.
$\textbf{procedure }\textsc{M}(x)$
$\qquad\textbf{if }0\leq x\leq 1$
$\qquad\qquad\textbf{return }x$
$\qquad\textbf{return }\textsc{M}(x^2\bmod 2^{32})$
Let $f:\mathbb N\to\mathbb N$ be defined such that $f(x) = 0$ if $\textsc{M}(x)$ does not terminate, and otherwise $f(x)$ equals the number of calls made to $\textsc{M}$ during the running of $\textsc{M}(x)$, not including the initial call. For example, $f(1) = 0$ and $f(2^{31}) = 1$. Compute the number of ones in the binary expansion of
\[
f(0) + f(1) + f(2) + \cdots + f(2^{32} - 1).
\]
2024 Poland - Second Round, 5
The positive reals $a, b, c, x, y, z$ satisfy $$5a+4b+3c=5x+4y+3z.$$ Show that $$\frac{a^5}{x^4}+\frac{b^4}{y^3}+\frac{c^3}{z^2} \geq x+y+z.$$
[i]Proposed by Dominik Burek[/i]
2021 Dutch IMO TST, 2
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
2003 AMC 10, 21
Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
$ \textbf{(A)}\ 22 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 27 \qquad
\textbf{(D)}\ 28 \qquad
\textbf{(E)}\ 29$