Found problems: 85335
2007 Stars of Mathematics, 1
Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies:
$ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $
$ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $
$ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $
2009 Tuymaada Olympiad, 4
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.
[i]Proposed by A. Khabrov[/i]
2004 Romania Team Selection Test, 6
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
1984 All Soviet Union Mathematical Olympiad, 382
Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\
y^2/ 3 + z^2 = 9 \\
z^2 + zx + x^2 = 16 \end{cases}$
Find the value of expression $xy + 2yz + 3zx$.
2019 Iran MO (3rd Round), 1
Given a number $k\in \mathbb{N}$. $\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{1,2,\cdots,9\}$. For all $n\geq 0$
$$\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .$$
Prove that there is a number $1\leq t \leq 9$ and $N\in \mathbb{N}$ such that $b_n=ta_n$ for all $n\geq N$.\\
(Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$)
2016 Fall CHMMC, 1
We say that the string $d_kd_{k-1} \cdots d_1d_0$ represents a number $n$ in base $-2$ if each $d_i$ is either $0$ or $1$,
and $n = d_k(-2)^k + d_{k-1}(-2)^{k-1} + \cdots + d_1(-2) + d_0$. For example, $110_{-2}$ represents the number $2$. What string represents $2016$ in base $-2$?
1965 Miklós Schweitzer, 2
Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).
2008 Junior Balkan Team Selection Tests - Romania, 2
In a sequence of natural numbers $ a_1,a_2,...,a_n$ every number $ a_k$ represents sum of the multiples of the $ k$ from sequence. Find all possible values for $ n$.
2001 Stanford Mathematics Tournament, 4
For what values of $a$ does the system of equations
\[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?
2021 JBMO TST - Turkey, 8
$w_1$ and $w_2$ circles have different diameters and externally tangent to each other at $X$. Points $A$ and $B$ are on $w_1$, points $C$ and $D$ are on $w_2$ such that $AC$ and $BD$ are common tangent lines of these two circles. $CX$ intersects $AB$ at $E$ and $w_1$ at $F$ second time. $(EFB)$ intersects $AF$ at $G$ second time. If $AX \cap CD =H$, show that points $E, G, H$ are collinear.
2009 Harvard-MIT Mathematics Tournament, 2
Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.
2000 239 Open Mathematical Olympiad, 6
$n$ cockroaches are sitting on the plane at the vertices of the regular $ n $ -gon. They simultaneously begin to move at a speed of $ v $ on the sides of the polygon in the direction of the clockwise adjacent cockroach, then they continue moving in the initial direction with the initial speed. Vasya a entomologist moves on a straight line in the plane at a speed of $u$. After some time, it turned out that Vasya has crushed three cockroaches. Prove that $ v = u $.
2018 South East Mathematical Olympiad, 6
In the isosceles triangle $ABC$ with $AB=AC$, the center of $\odot O$ is the midpoint of the side $BC$, and $AB,AC$ are tangent to the circle at points $E,F$ respectively. Point $G$ is on $\odot O$ with $\angle AGE = 90^{\circ}$. A tangent line of $\odot O$ passes through $G$, and meets $AC$ at $K$. Prove that line $BK$ bisects $EF$.
2007 Federal Competition For Advanced Students, Part 1, 1
In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field.
For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column.
$ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$.
Show that $ A\plus{}B\neq0$
1996 Tournament Of Towns, (514) 1
Consider three edges $a, b, c$ of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from $a$, $b$ and $c$.
(V Proizvolov,)
2021 AMC 12/AHSME Fall, 3
Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?
$\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\
5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$
2020 ABMC, Team
[u]Round 1[/u]
[b]1.1.[/b] A person asks for help every $3$ seconds. Over a time period of $5$ minutes, how many times will they ask for help?
[b]1.2.[/b] In a big bag, there are $14$ red marbles, $15$ blue marbles, and$ 16$ white marbles. If Anuj takes a marble out of the bag each time without replacement, how many marbles does Anuj need to remove to be sure that he will have at least $3$ red marbles?
[b]1.3.[/b] If Josh has $5$ distinct candies, how many ways can he pick $3$ of them to eat?
[u]Round 2[/u]
[b]2.1.[/b] Annie has a circular pizza. She makes $4$ straight cuts. What is the minimum number of slices of pizza that she can make?
[b]2.2.[/b] What is the sum of the first $4$ prime numbers that can be written as the sum of two perfect squares?
[b]2.3.[/b] Consider a regular octagon $ABCDEFGH$ inscribed in a circle of area $64\pi$. If the length of arc $ABC$ is $n\pi$, what is $n$?
[u]Round 3[/u]
[b]3.1.[/b] Let $ABCDEF$ be an equiangular hexagon with consecutive sides of length $6, 5, 3, 8$, and $3$. Find the length of the sixth side.
[b]3.2.[/b] Jack writes all of the integers from $ 1$ to $ n$ on a blackboard except the even primes. He selects one of the numbers and erases all of its digits except the leftmost one. He adds up the new list of numbers and finds that the sum is $2020$. What was the number he chose?
[b]3.3.[/b] Our original competition date was scheduled for April $11$, $2020$ which is a Saturday. The numbers $4116$ and $2020$ have the same remainder when divided by $x$. If $x$ is a prime number, find the sum of all possible $x$.
[u]Round 4[/u]
[b]4.1.[/b] The polynomials $5p^2 + 13pq + cq^2$ and $5p^2 + 13pq - cq^2$ where $c$ is a positive integer can both be factored into linear binomials with integer coefficients. Find $c$.
[b]4.2.[/b] In a Cartesian coordinate plane, how many ways are there to get from $(0, 0)$ to $(2, 3)$ in $7$ moves, if each move consists of a moving one unit either up, down, left, or right?
[b]4.3.[/b] Bob the Builder is building houses. On Monday he finds an empty field. Each day starting on Monday, he finishes building a house at noon. On the $n$th day, there is a $\frac{n}{8}$ chance that a storm will appear at $3:14$ PM and destroy all the houses on the field. At any given moment, Bob feels sad if and only if there is exactly $1$ house left on the field that is not destroyed. The probability that he will not be sad on Friday at $6$ PM can be expressed as $p/q$ in simplest form. Find $p + q$.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784570p24468605]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 HMNT, 14
Suppose that point $D$ lies on side $BC$ of triangle $ABC$ such that $AD$ bisects $\angle BAC,$ and let $\ell$ denote the line through $A$ perpendicular to $AD.$ If the distances from $B$ and $C$ to $\ell$ are $5$ and $6,$ respectively, compute $AD.$
2016 PUMaC Algebra Individual A, A6
Let $[a, b] = ab - a - b$. Shaq sees the numbers $2, 3, \dots , 101$ written on a blackboard. Let $V$ be the largest number that Shaq can obtain by repeatedly choosing two numbers $a, b$ on the board and replacing them with $[a, b]$ until there is only one number left. Suppose $N$ is the integer with $N!$ nearest to $V$. Find the nearest integer to $10^6 \cdot \tfrac{|V-N!|}{N!}$.
2007 Junior Balkan Team Selection Tests - Romania, 1
Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.
1984 Tournament Of Towns, (055) O3
Consider the $4(N-1)$ squares on the boundary of an $N$ by $N$ array of squares. We wish to insert in these squares $4 (N-1)$ consecutive integers (not necessarily positive) so that the sum of the numbers at the four vertices of any rectangle with sides parallel to the diagonals of the array (in the case of a “degenerate” rectangle, i.e. a diagonal, we refer to the sum of the two numbers in its corner squares) are one and the same number.
Is this possible? Consider the cases
(a) $N = 3$
(b) $N = 4$
(c) $N = 5$
(VG Boltyanskiy, Moscow)
2019 Mathematical Talent Reward Programme, MCQ: P 4
Suppose $\triangle ABC$ is a triangle. From the vertex $A$ draw the altitude $AH$, angle bisector (of $\angle BAC$) $AP$, median $AD$ and these intersect the side $BC$ at the points (from left in order) $H$, $P$, $D$ respectively. Let $\angle CAH = \angle HAP = \angle PAD = \angle DAB$. Then $\angle ACH =$
[list=1]
[*] $22.5^{\circ}$
[*] $45^{\circ}$
[*] $67.5^{\circ}$
[*] None of the above
[/list]
1999 Romania Team Selection Test, 17
A polyhedron $P$ is given in space. Find whether there exist three edges in $P$ which can be the sides of a triangle. Justify your answer!
[i]Barbu Berceanu[/i]
2019 Caucasus Mathematical Olympiad, 8
Determine if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_{101}$, $b_1$, $b_2$, \ldots, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,101\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 100!+\sum\limits_{i\in S}b_i \right)$.
2021 Swedish Mathematical Competition, 3
Four coins are laid out on a table so that they form the corners of a square. One move consists of tipping one of the coins by letting it jump over one of the others the coin so that it ends up on the directly opposite side of the other coin, the same distance from as it was before the move was made. Is it possible to make a number of moves so that the coins ends up in the corners of a square with a different side length than the original square?