Found problems: 85335
2003 AIME Problems, 4
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)$, find $n$.
LMT Team Rounds 2021+, 6
For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.
2023 CMWMC, R6
[b]p16.[/b] Let $P(x)$ be a quadratic such that $P(-2) = 10$, $P(0) = 5$, $P(3) = 0$. Then, find the sum of the coefficients of the polynomial equal to $P(x)P(-x)$.
[b]p17.[/b] Suppose that $a < b < c < d$ are positive integers such that the pairwise differences of $a, b, c, d$ are all distinct, and $a + b + c + d$ is divisible by $2023$. Find the least possible value of $d$.
[b]p18.[/b] Consider a right rectangular prism with bases $ABCD$ and $A'B'C'D'$ and other edges $AA'$, $BB'$, $CC'$ and $DD'$. Suppose $AB = 1$, $AD = 2$, and $AA' = 1$.
$\bullet$ Let $X$ be the plane passing through $A$, $C'$, and the midpoint of $BB'$.
$\bullet$ Let $Y$ be the plane passing through $D$, $B'$, and the midpoint of $CC'$.
Then the intersection of $X$, $Y$ , and the prism is a line segment of length $\ell$. Find $\ell$.
PS. You should use hide for answers.
2020 Purple Comet Problems, 19
Right $\vartriangle ABC$ has side lengths $6, 8$, and $10$. Find the positive integer $n$ such that the area of the region inside the circumcircle but outside the incircle of $\vartriangle ABC$ is $n\pi$.
[img]https://cdn.artofproblemsolving.com/attachments/d/1/cb112332069c09a3b370343ca8a2ef21102fe2.png[/img]
2020 CCA Math Bonanza, I12
Find all pairs $(a,b)$ of positive integers satisfying the following conditions:
- $a\leq b$
- $ab$ is a perfect cube
- No divisor of $a$ or $b$ is a perfect cube greater than $1$
- $a^2+b^2=85\text{lcm}(a,b)$
[i]2020 CCA Math Bonanza Individual Round #12[/i]
2002 Argentina National Olympiad, 5
Let $\vartriangle ABC$ be an isosceles triangle with $AC = BC$. Points $D, E, F$ are considered on $BC, CA, AB$, respectively, such that $AF> BF$ and that the quadrilateral $CEFD$ is a parallelogram. The perpendicular line to $BC$ drawn by $B$ intersects the perpendicular bisector of $AB$ at $G$. Prove that $DE \perp FG$.
2007 AIME Problems, 10
Let $S$ be a set with six elements. Let $P$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $P$. the probability that $B$ is contained in at least one of $A$ or $S-A$ is $\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive integers, $n$ is prime, and $m$ and $n$ are relatively prime. Find $m+n+r.$ (The set $S-A$ is the set of all elements of $S$ which are not in $A.$)
2005 Iran MO (3rd Round), 1
Suppose $a,b,c\in \mathbb R^+$. Prove that :\[\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\]
MIPT student olimpiad spring 2023, 1
In $R^n$ is given $n-1$ vectors, the coordinates of each are zero-sum integers.
Prove that the $(n-1)$-dimensional volume of an $(n-1)$-dimensional parallelepiped $P$ stretched
by these vectors, is the product of an integer and $\sqrt(n)$.
2009 Princeton University Math Competition, 3
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.
2022 Iran MO (3rd Round), 2
$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.
1980 AMC 12/AHSME, 29
How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below?
\[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \]
$\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$
2011 Kosovo National Mathematical Olympiad, 2
Find all solutions to the equation:
\[ \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16 \]
2021 Princeton University Math Competition, B2
Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for $n$ minutes once he arrives at the airport before going through security. What is the smallest $n$ they can select such that they meet with at least 50% probability? The answer will be of the form $a + b\sqrt{c}$ for integers $a$, $b$, and $c$, where $c$ has no perfect square factor other than $1$. Report $a + b + c.$
2013 JBMO Shortlist, 4
A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates.
a) Find a latticed rectangle of area $2013$, whose sides are not parallel to the axes.
b) Show that if a latticed rectangle has area $2011$, then their sides are parallel to the axes.
2019 Serbia National MO, 4
For a $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1 $ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $ AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.
1983 IMO Shortlist, 19
Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
\[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\]
Prove that $P(1983) = F_{1983} - 1.$
2007 Harvard-MIT Mathematics Tournament, 2
Two reals $x$ and $y$ are such that $x-y=4$ and $x^3-y^3=28$. Compute $xy$.
1991 Irish Math Olympiad, 5
Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties:
(a) $0$ is not in $S$;
(b) for each $s_1,s_2$ in $S$, the rational number $s_1/s_2$ is in $S$;
(c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$.
Prove that if $x$ belongs to $S$, then there exists elements $y,z$ in $S$ such that $x=y+z$.
2021 BMT, 24
Given that $x, y$, and $z$ are a combination of positive integers such that $xyz = 2(x + y + z)$, compute the sum of all possible values of $x + y + z$.
1945 Moscow Mathematical Olympiad, 105
A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.
1995 Belarus National Olympiad, Problem 6
Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root.
2019 Saudi Arabia BMO TST, 1
Let $p$ be an odd prime number.
a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n.
b) Find all $n$ satisfy condition above when $p = 3$
1991 Tournament Of Towns, (308) 5
A $9 \times 9$ square is divided into $81$ unit cells. Some of the cells are coloured. The distance between the centres of any two coloured cells is more than $2$.
(a) Give an example of colouring with $17$ coloured cells.
(b) Prove that the numbers of coloured cells cannot exceed $17$.
(S. Fomin, Leningrad)
2024 Putnam, B4
Let $n$ be a positive integer. Set $a_{n,0}=1$. For $k\geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\{1,\,\ldots,\,n\}$, and let
\[
a_{n,k+1}=
\begin{cases}
a_{n,k}+1, & \text{if $m_{n,k}>a_{n,k}$;}\\
a_{n,k}, & \text{if $m_{n,k}=a_{n,k}$;}\\
a_{n,k}-1, & \text{if $m_{n,k}<a_{n,k}$.}
\end{cases}
\]
Let $E(n)$ be the expected value of $a_{n,n}$. Determine $\lim_{n\to\infty}E(n)/n$.