Found problems: 85335
2010 Purple Comet Problems, 7
Find the sum of the digits in the decimal representation of the number $5^{2010} \cdot 16^{502}.$
1957 AMC 12/AHSME, 43
We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $ x$-axis, the line $ x \equal{} 4$, and the parabola $ y \equal{} x^2$ is:
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 35\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 30\qquad
\textbf{(E)}\ \text{not finite}$
1988 IMO Longlists, 51
The positive integer $n$ has the property that, in any set of $n$ integers, chosen from the integers $1,2, \ldots, 1988,$ twenty-nine of them form an arithmetic progression. Prove that $n > 1788.$
2020 AMC 8 -, 23
Five different awards are to be given to three students. Each student will receive at least one award. In how many ways can the awards be distributed?
$\textbf{(A)}\ 120 \qquad \textbf{(B)}\ 150 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 210 \qquad \textbf{(E)}\ 240$
Mathematical Minds 2024, P4
Let $a$, $b$, $c$ be positive real numbers such that $a+b+c=3$. Prove that $$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+a^3}{2}}\leqslant a^2+b^2+c^2.$$
[i]Proposed by Andrei Vila[/i]
2013 Online Math Open Problems, 16
Al has the cards $1,2,\dots,10$ in a row in increasing order. He first chooses the cards labeled $1$, $2$, and $3$, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled $2$, $3$, and $4$, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence $3,2,1,4,5,\dots,$ and his second move could have rearranged that to $2,4,1,3,5,\dots$.) He continues this process until he has rearranged the cards with labels $8$, $9$, $10$. Determine the number of possible orderings of cards he can end up with.
[i]Proposed by Ray Li[/i]
2020 Purple Comet Problems, 24
Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt2$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.
2002 India IMO Training Camp, 19
Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that
\[
\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad
\angle CFB = 2 \angle ACB.
\]
Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum
\[
\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.
\]
2009 Today's Calculation Of Integral, 519
Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.
2007 VJIMC, Problem 3
A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)
2022-IMOC, G6
Let $D$ be a point on the circumcircle of some triangle $ABC$. Let $E, F$ be points on $AC$, $AB$, respectively, such that $A,D,E,F$ are concyclic. Let $M$ be the midpoint of $BC$. Show that if $DM$, $BE$, $CF$ are concurrent, then either $BE \cap CF$ is on the circle $ADEF$, or $EF$ is parallel to $BC$.
[i]proposed by USJL[/i]
1974 IMO Longlists, 21
Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+.$
1997 Belarusian National Olympiad, 1
$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.
2018 India PRMO, 22
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
2012 Dutch Mathematical Olympiad, 4
We are given an acute triangle $ABC$ and points $D$ on $BC$ and $E$ on $AC$ such that $AD$ is perpendicular to $BC$ and $BE$ is perpendicular to $AC$. The intersection of $AD$ and $BE$ is called $H$. A line through $H$ intersects line segment $BC$ in $P$, and intersects line segment $AC$ in $Q$. Furthermore, $K$ is a point on $BE$ such that $PK$ is perpendicular to $BE$, and $L$ is a point on $AD$ such that $QL$ is perpendicular to $AD$. Prove that $DK$ and $EL$ are parallel.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, H, K, L, P, Q;
A = (0,0);
B = (6,0);
C = (3.5,4);
D = (A + reflect(B,C)*(A))/2;
E = (B + reflect(A,C)*(B))/2;
H = extension(A, D, B, E);
P = extension(H, H + dir(-10), B, C);
Q = extension(H, H + dir(-10), A, C);
K = (P + reflect(B,E)*(P))/2;
L = (Q + reflect(A,D)*(Q))/2;
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(K--P--Q--L);
draw(rightanglemark(B,D,A,5));
draw(rightanglemark(B,E,A,5));
draw(rightanglemark(P,K,B,5));
draw(rightanglemark(A,L,Q,5));
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, N);
dot("$D$", D, NE);
dot("$E$", E, NW);
dot("$H$", H, N);
dot("$K$", K, SW);
dot("$L$", L, SE);
dot("$P$", P, NE);
dot("$Q$", Q, NW);
[/asy]
2014 Stanford Mathematics Tournament, 6
Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribe an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is parallel to the major axis. Compute the area of $\vartriangle ABC$.
2003 Estonia National Olympiad, 2
Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.
1983 AMC 12/AHSME, 2
Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3 \text{cm}$ from $P$?
$\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 8$
2005 National Olympiad First Round, 16
$100$ stones, each weighs $1$ kg or $10$ kgs or $50$ kgs, weighs $500$ kgs in total. How many values can the number of stones weighing $10$ kgs take?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
1993 China National Olympiad, 6
Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.
2014 Indonesia MO Shortlist, C3
Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.
1996 All-Russian Olympiad, 8
The numbers from 1 to 100 are written in an unknown order. One may ask about any 50 numbers and find out their relative order. What is the fewest questions needed to find the order of all 100 numbers?
[i]S. Tokarev[/i]
LMT Team Rounds 2010-20, A9
$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$.
[i]Proposed by Alex Li[/i]
2018 Mathematical Talent Reward Programme, SAQ: P 6
Let $d(n)$ be the number of divisors of $n,$ where $n$ is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence $\left\{a_{1}, a_{2}, \cdots\right\}$ if $\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\}$ is an nonconstant geometric series then $\left\{a_{1}, a_{2}, \cdots\right\}$ does not bear same colour.
1985 Putnam, A3
Let $d$ be a real number. For each integer $m \geq 0,$ define a sequence $\left\{a_{m}(j)\right\}, j=0,1,2, \ldots$ by the condition
\begin{align*}
a_{m}(0)&=d / 2^{m},\\
a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j), \quad j \geq 0.
\end{align*}
Evaluate $\lim _{n \rightarrow \infty} a_{n}(n).$