Found problems: 85335
2009 Today's Calculation Of Integral, 421
Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions.
(1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$.
(2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal.
(3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.
2005 Flanders Junior Olympiad, 3
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
1998 Slovenia National Olympiad, Problem 3
A point $A$ is outside a circle $\mathcal K$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle in $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ and $F$ and line $AD$ at $D$ and $G$. Prove that the lines $BD$ and $FG$ are parallel.
KoMaL A Problems 2017/2018, A. 719
Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. The $A$-excircle, $B$-excircle, and $C$-excircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$. Show that if $M$ is the midpoint of $PA_1$, then lines $HM$ and $OI$ are parallel.
[i]Michael Ren[/i]
2018 BMT Spring, 4
Alice starts with an empty string and randomly appends one of the digits $2$, $0$, $1$, or $8$ until the string ends with $2018$. What is the probability Alice appends less than $9$ digits before stopping?
1997 All-Russian Olympiad, 3
Find all triples $m$; $n$; $l$ of natural numbers such that
$m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$:
[i]S. Tokarev[/i]
2017 Princeton University Math Competition, A8
Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$, then $N = 7$.
2020 Spain Mathematical Olympiad, 1
A polynomial $p(x)$ with real coefficients is said to be [i]almeriense[/i] if it is of the form:
$$
p(x) = x^3+ax^2+bx+a
$$
And its three roots are positive real numbers in arithmetic progression. Find all [i]almeriense[/i] polynomials such that $p\left(\frac{7}{4}\right) = 0$
2020 Online Math Open Problems, 19
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that
[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]
[i]Proposed by Jaedon Whyte[/i]
2011 Flanders Math Olympiad, 2
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.
2021 Yasinsky Geometry Olympiad, 5
In triangle $ABC$, point $I$ is the center of the inscribed circle. $AT$ is a segment tangent to the circle circumscribed around the triangle $BIC$ . On the ray $AB$ beyond the point$ B$ and on the ray $AC$ beyond the point $C$, we draw the segments $BD$ and $CE$, respectively, such that $BD = CE = AT$. Let the point $F$ be such that $ABFC$ is a parallelogram. Prove that points $D, E$ and $F$ lie on the same line.
(Dmitry Prokopenko)
2019 HMNT, 2
$2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible number of items one student can receive, over all possible values of $N$ and all possible ways of voting.
1991 AMC 8, 10
The area in square units of the region enclosed by parallelogram $ABCD$ is
[asy]
unitsize(24);
pair A,B,C,D;
A=(-1,0); B=(0,2); C=(4,2); D=(3,0);
draw(A--B--C--D); draw((0,-1)--(0,3)); draw((-2,0)--(6,0));
draw((-.25,2.75)--(0,3)--(.25,2.75)); draw((5.75,.25)--(6,0)--(5.75,-.25));
dot(origin); dot(A); dot(B); dot(C); dot(D); label("$y$",(0,3),N); label("$x$",(6,0),E);
label("$(0,0)$",origin,SE); label("$D (3,0)$",D,SE); label("$C (4,2)$",C,NE);
label("$A$",A,SW); label("$B$",B,NW);
[/asy]
$\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 18$
2007 ITest, 1
A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.)
$\textbf{(A) }4$
2023 Korea - Final Round, 2
Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition.
(Condition) For each positive real number $x$, there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$, and the number of $y$ is finite.
Prove $f(x) > f(y)$ for any positive real numbers $x < y$. ($\mathbb{R^+}$ is a set for all positive real numbers.)
2008 Estonia Team Selection Test, 5
Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.
2006 All-Russian Olympiad Regional Round, 10.7
For what positive integers $n$ are there positive rational, but not integer, numbers $a$ and $b$ such that both numbers $a + b$ and $a^n + b^n$ are integers?
2021 Balkan MO Shortlist, N2
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively
defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there
exists some $i \in \mathbb{N}$ with $a_i = m^2$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
2000 AMC 12/AHSME, 20
If $ x$, $ y$, and $ z$ are positive numbers satisfying \[x \plus{} 1/y \equal{} 4,\quad y \plus{} 1/z \equal{} 1,\quad\text{and}\quad z \plus{} 1/x \equal{} 7/3,\] then $ xyz \equal{}$
$ \textbf{(A)}\ 2/3 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 7/3$
2022 Macedonian Team Selection Test, Problem 1
Let $n$ be a fixed positive integer. There are $n \geq 1$ lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer $i$ ($1 \leq i \leq n$) and switch the state of the first $i$ lamps from the left. Determine the smallest number $k$ with the property that we can make all of the lamps be switched on using at most $k$ moves, no matter what the initial configuration was.
[i]Proposed by Viktor Simjanoski and Nikola Velov[/i]
2019 MIG, 12
Calculate the product $\tfrac13 \times \tfrac24 \times \tfrac35 \times \cdots \times \tfrac{18}{20} \times \tfrac{19}{21}$.
$\textbf{(A) }\dfrac{1}{210}\qquad\textbf{(B) }\dfrac{1}{190}\qquad\textbf{(C) }\dfrac{1}{21}\qquad\textbf{(D) }\dfrac{1}{20}\qquad\textbf{(E) }\dfrac{1}{10}$
1991 All Soviet Union Mathematical Olympiad, 540
$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.
2020 Estonia Team Selection Test, 1
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$.
Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .
2001 Moldova National Olympiad, Problem 3
A line $d_i~(i=1,2,3)$ intersects two opposite sides of a square $ABCD$ at points $M_i$ and $N_i$. Prove that if $M_1N_1=M_2N_2=M_3N_3$, then two of the lines $d_i$ are either parallel or perpendicular.
Estonia Open Senior - geometry, 1993.5
Within an equilateral triangle $ABC$, take any point $P$. Let $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.