Found problems: 85335
1954 AMC 12/AHSME, 22
The expression $ \frac{2x^2-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}$ cannot be evaluated for $ x=-1$ or $ x=2$, since division by zero is not allowed. For other values of $ x$:
$\textbf{(A)}\ \text{The expression takes on many different values.} \\
\textbf{(B)}\ \text{The expression has only the value 2.} \\
\textbf{(C)}\ \text{The expression has only the value 1.} \\
\textbf{(D)}\ \text{The expression always has a value between } -1 \text{ and } +2. \\
\textbf{(E)}\ \text{The expression has a value greater than 2 or less than } -1.$
2010 Baltic Way, 7
There are some cities in a country; one of them is the capital. For any two cities $A$ and $B$ there is a direct flight from $A$ to $B$ and a direct flight from $B$ to $A$, both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.
2010 Stanford Mathematics Tournament, 18
In an $n$-by-$m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$
2015 ASDAN Math Tournament, 1
Rio likes fruit, and one day she decides to pick persimmons. She picks a total of $12$ persimmons from the first $5$ trees she sees. Rio has $5$ more trees to pick persimmons from. If she wants to pick an average of $4$ persimmons per tree overall, what is the average number of persimmons that she must pick from each of the last $5$ trees for her goal?
2023 Abelkonkurransen Finale, 3b
Find all integers $a$ and $b$ satisfying
\begin{align*}
a^6 + 1 & \mid b^{11} - 2023b^3 + 40b, \qquad \text{and}\\
a^4 - 1 & \mid b^{10} - 2023b^2 - 41.
\end{align*}
2001 Kazakhstan National Olympiad, 1
Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.
2015 Switzerland Team Selection Test, 2
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
2014 India IMO Training Camp, 2
Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.
2002 China Western Mathematical Olympiad, 3
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.
1982 Swedish Mathematical Competition, 3
Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.
2019 IFYM, Sozopol, 7
Let $G$ be a bipartite graph in which the greatest degree of a vertex is 2019. Let $m$ be the least natural number for which we can color the edges of $G$ in $m$ colors so that each two edges with a common vertex from $G$ are in different colors. Show that $m$ doesn’t depend on $G$ and find its value.
STEMS 2021 Math Cat A, Q2
Suppose $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ is a function such that $\frac{f(x)}{x}$ is increasing on $\mathbb{R}^{+}$. For $a,b,c>0$, prove that $$2\left (\frac{f(a)+f(b)}{a+b} + \frac{f(b)+f(c)}{b+c}+ \frac{f(c)+f(a)}{c+a} \right) \geq 3\left(\frac{f(a)+f(b)+f(c)}{a+b+c}\right) + \frac{f(a)}{a}+ \frac{f(b)}{b}+ \frac{f(c)}{c}$$
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle. The inside bisector of the angle $\angle BAC$ cuts $[BC]$ in $L$ and the circle $(C)$ circumsbribed to the triangle $ABC$ in $D$. The perpendicular to $(AC)$ going through $D$ cuts $[AC]$ in $M$ and the circle $(C)$ in $K$. Find the value of $\frac{AM}{MC}$ knowing that $\frac{BL}{LC}=\frac{1}{2}$.
MBMT Guts Rounds, 2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]B16 / G11[/b] Let triangle $ABC$ be an equilateral triangle with side length $6$. If point $D$ is on $AB$ and point $E$ is on $BC$, find the minimum possible value of $AD + DE + CE$.
[b]B17 / G12[/b] Find the smallest positive integer $n$ with at least seven divisors.
[b]B18 / G13[/b] Square $A$ is inscribed in a circle. The circle is inscribed in Square $B$. If the circle has a radius of $10$, what is the ratio between a side length of Square $A$ and a side length of Square $B$?
[b]B19 / G14[/b] Billy Bob has $5$ distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other?
[b]B20 / G15[/b] Six people make statements as follows:
Person $1$ says “At least one of us is lying.”
Person $2$ says “At least two of us are lying.”
Person $3$ says “At least three of us are lying.”
Person $4$ says “At least four of us are lying.”
Person $5$ says “At least five of us are lying.”
Person $6$ says “At least six of us are lying.”
How many are lying?
[u]Set 5[/u]
[b]B21 / G16[/b] If $x$ and $y$ are between $0$ and $1$, find the ordered pair $(x, y)$ which maximizes $(xy)^2(x^2 - y^2)$.
[b]B22 / G17[/b] If I take all my money and divide it into $12$ piles, I have $10$ dollars left. If I take all my money and divide it into $13$ piles, I have $11$ dollars left. If I take all my money and divide it into $14$ piles, I have $12$ dollars left. What’s the least amount of money I could have?
[b]B23 / G18[/b] A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation.
[b]B24 / G20[/b] A regular $12$-sided polygon is inscribed in a circle. Gaz then chooses $3$ vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right?
[b]B25 / G22[/b] A book has at most $7$ chapters, and each chapter is either $3$ pages long or has a length that is a power of $2$ (including $1$). What is the least positive integer $n$ for which the book cannot have $n$ pages?
[u]Set 6[/u]
[b]B26 / G26[/b] What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers?
[b]B27 / G27[/b] Estimate $12345^{\frac{1}{123}}$.
[b]B28 / G28[/b] Let $O$ be the center of a circle $\omega$ with radius $3$. Let $A, B, C$ be randomly selected on $\omega$. Let $M$, $N$ be the midpoints of sides $BC$, $CA$, and let $AM$, $BN$ intersect at $G$. What is the probability that $OG \le 1$?
[b]B29 / G29[/b] Let $r(a, b)$ be the remainder when $a$ is divided by $b$. What is $\sum^{100}_{i=1} r(2^i , i)$?
[b]B30 / G30[/b] Bongo flips $2023$ coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets $HHHT T HT T HHHHT H$, he’d have maximal runs of length $3, 1, 4, 1$. Bongo squares the lengths of all his maximal runs and adds them to get a number $M$. What is the expected value of $M$?
- - - - - -
[b]G19[/b] Let $ABCD$ be a square of side length $2$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. Let the intersection of $BN$ and $CM$ be $E$. Find the area of quadrilateral $NECD$.
[b]G21[/b] Quadrilateral $ABCD$ has $\angle A = \angle D = 60^o$. If $AB = 8$, $CD = 10$, and $BC = 3$, what is length $AD$?
[b]G23[/b] $\vartriangle ABC$ is an equilateral triangle of side length $x$. Three unit circles $\omega_A$, $\omega_B$, and $\omega_C$ lie in the plane such that $\omega_A$ passes through $A$ while $\omega_B$ and $\omega_C$ are centered at $B$ and $C$, respectively. Given that $\omega_A$ is externally tangent to both $\omega_B$ and $\omega_C$, and the center of $\omega_A$ is between point $A$ and line $\overline{BC}$, find $x$.
[b]G24[/b] For some integers $n$, the quadratic function $f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12)$ has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form $2^k$ for some nonnegative integer $k$. What is the sum of all possible values of $n$?
[b]G25[/b] In a triangle, let the altitudes concur at $H$. Given that $AH = 30$, $BH = 14$, and the circumradius is $25$, calculate $CH$
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2005 Today's Calculation Of Integral, 88
A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$
Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$
2008 Iran MO (3rd Round), 5
Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$
\[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]
1982 Miklós Schweitzer, 4
Let \[ f(n)= \sum_{p|n , \;p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .\] Prove that \[ \limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .\]
[i]P. Erdos[/i]
1995 AMC 12/AHSME, 24
There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that \[A \log_{200} 5 + B \log_{200} 2 = C. \] What is $A+B+C$?
$\textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$
2013 Tournament of Towns, 5
In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.
2024 OMpD, 1
We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties:
1. \( T \) has at least two distinct elements;
2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \).
For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \).
What is the largest possible number of elements that a [b]kawaii [/b]subset can have?
2023 CUBRMC, 8
If $r$ is real number sampled at random with uniform probability, find the probability that $r$ is [i]strictly [/i] closer to a multiple of $58$ than it is to a multiple of $37$.
1981 Romania Team Selection Tests, 4.
Let $n\geqslant 3$ be a fixed integer and $\omega=\cos\dfrac{2\pi}n+i\sin\dfrac{2\pi}n$.
Show that for every $a\in\mathbb{C}$ and $r>0$, the number
\[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\]
is an integer. Interpet this result geometrically.
[i]Octavian Stănășilă[/i]
2011 USA Team Selection Test, 7
Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.
2022 IFYM, Sozopol, 5
Prove that
$\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.
2022 Yasinsky Geometry Olympiad, 2
In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$.
(Gryhoriy Filippovskyi)