Found problems: 85335
2022-2023 OMMC, 6
Find the unique integer $\overline{CA7DB}$ with nonzero digits so that $\overline{ABCD} \cdot 3 = \overline{CA7DB}.$
2014 Math Prize For Girls Problems, 4
Say that an integer $A$ is [i]yummy[/i] if there exist several consecutive integers (including $A$) that add up to 2014. What is the smallest yummy integer?
2019 CMIMC, 3
Let $ABC$ be an equilateral triangle with side length $2$, and let $M$ be the midpoint of $\overline{BC}$. Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$. What is the length of $\overline{XY}$?
2012 China Girls Math Olympiad, 7
Let $\{a_n\}$ be a sequence of nondecreasing positive integers such that $\textstyle\frac{r}{a_r} = k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that $\textstyle\frac{s}{a_s} = k$.
1970 AMC 12/AHSME, 13
Given the binary operation $\ast$ defined by $a\ast b=a^b$ for all positive numbers $a$ and $b$. The for all positive $a,b,c,n,$ we have
$\textbf{(A) }a\ast b=b\ast a\qquad\textbf{(B) }a\ast (b\ast c)=(a\ast b)\ast c\qquad$
$\textbf{(C) }(a\ast b^n)=(a\ast n)\ast b\qquad\textbf{(D) }(a\ast b)^n=a\ast (bn)\qquad \textbf{(E) }\text{None of these}$
2011 Federal Competition For Advanced Students, Part 2, 1
Determine all pairs $(a,b)$ of non-negative integers, such that $a^b+b$ divides $a^{2b}+2b$.
(Remark: $0^0=1$.)
2005 AMC 12/AHSME, 3
A rectangle with a diagonal of length $ x$ is twice as long as it is wide. What is the area of the rectangle?
$ \textbf{(A)}\ \frac14x^2 \qquad
\textbf{(B)}\ \frac25x^2 \qquad
\textbf{(C)}\ \frac12x^2 \qquad
\textbf{(D)}\ x^2 \qquad
\textbf{(E)}\ \frac32x^2$
1989 Tournament Of Towns, (215) 3
Find six distinct positive integers such that the product of any two of them is divisible by their sum.
(D. Fomin, Leningrad)
2014 BMT Spring, 1
Find all real numbers $x$ such that $4^x-2^{x+2}+3=0$.
2024 May Olympiad, 1
A $4\times 8$ grid is divided into $32$ unit squares. There are square tiles of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. The goal is to completely cover the grid using exactly $n$ of these tiles.
[list=a]
[*]Is it possible to do this if $n = 19$?
[*]Is it possible to do this if $n = 14$?
[*]Is it possible to do this if $n = 7$?
[/list]
[b]Note:[/b] The tiles cannot overlap or extend beyond the grid.
2014 Cuba MO, 3
Ana and Carlos entertain themselves with the next game. At the beginning of game in each vertex of the square there is an empty box. In each step, the corresponding player has two possibilities: either he adds a stone to an arbitrary box, or move each box clockwise to the next vertex of the square. Carlos starts and they take 2012 steps in turn (each player 1006). So Carlos marks one of the vertices of the square and allows Ana to make a more play. Carlos wins if after this last step the number ofstones in some box is greater than the number of stones in the box which is at the vertex marked by Carlos; otherwise Ana wins. Which of the two players has a winning strategy?
2023 OMpD, 2
Let $ABCDE$ be a convex pentagon inscribed in a circle $\Gamma$, such that $AB = BC = CD$. Let $F$ and $G$ be the intersections of $BE$ with $AC$ and of $CE$ with $BD$, respectively. Show that:
a) $[ABC] = [FBCG]$
b) $\frac{[EFG]}{[EAD]} = \frac{BC}{AD}$
[b]Note: [/b] $[X]$ denotes the area of polygon $X$.
2024 Serbia JBMO TST, 3
a) Is it possible to place $2024$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains even number of checkers?
b) Is it possible to place $2023$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains odd number of checkers?
1985 All Soviet Union Mathematical Olympiad, 400
The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.
2018-2019 Winter SDPC, 4
Tom is chasing Jerry on the coordinate plane. Tom starts at $(x, y)$ and Jerry starts at $(0, 0)$. Jerry moves to the right at $1$ unit per second. At each positive integer time $t$, if Tom is within $1$ unit of Jerry, he hops to Jerry’s location and catches him. Otherwise, Tom hops to the midpoint of his and Jerry’s location.
[i]Example. If Tom starts at $(3, 2)$, then at time $t = 1$ Tom will be at $(2, 1)$ and Jerry will be at $(1, 0)$. At $t = 2$ Tom will catch Jerry.[/i]
Assume that Tom catches Jerry at some integer time $n$.
(a) Show that $x \geq 0$.
(b) Find the maximum possible value of $\frac{y}{x+1}$.
2015 Saint Petersburg Mathematical Olympiad, 6
There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible.
(D. Karpov)
2023 Junior Balkan Team Selection Tests - Moldova, 4
On the board there are three real numbers $(a,b,c)$. During a $procedure$ the numbers are erased and in their place another three numbers a written, either $(c,b,a)$ or every time a nonzero real number $ d $ is chosen and the numbers $(a, 2ad+b, ad^2+bd+c)$ are written.
1) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,0,-1)$ on the board after a finite number of procedures?
2) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,-1,-1)$ on the board after a finite number of procedures?
2005 Today's Calculation Of Integral, 13
Calculate the following integarls.
[1] $\int x\cos ^ 2 x dx$
[2] $\int \frac{x-1}{(3x-1)^2}dx$
[3] $\int \frac{x^3}{(2-x^2)^4}dx$
[4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$
[5] $\int (\ln x)^2 dx$
2005 Alexandru Myller, 1
Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation
$$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$
Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $
2017 Princeton University Math Competition, A1/B3
Triangle $ABC$ has $AB=BC=10$ and $CA=16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$.
2011 Silk Road, 2
Given an isosceles triangle $ABC$ with base $AB$. Point $K$ is taken on the extension of the side $AC$ (beyond the point $C$ ) so that $\angle KBC = \angle ABC$. Denote $S$ the intersection point of angle - bisectors of $\angle BKC$ and $\angle ACB$. Lines $AB$ and $KS$ intersect at point $L$, lines $BS$ and $CL$ intersect at point $M$ . Prove that line $KM$ passes through the midpoint of the segment $BC$.
2023 Indonesia TST, 1
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
2019 Putnam, B2
For all $n\ge 1$, let $a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}$. Determine $\lim_{n\rightarrow \infty}\frac{a_n}{n^3}$.
2003 Croatia National Olympiad, Problem 3
The natural numbers $1$ through $2003$ are arranged in a sequence. We repeatedly perform the following operation: If the first number in the sequence is $k$, the order of the first $k$ terms is reversed. Prove that after several operations number $1$ will occur on the first place.
2006 AMC 10, 1
Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
$ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$