Found problems: 85335
1979 IMO Longlists, 69
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
2019 Online Math Open Problems, 24
We define the binary operation $\times$ on elements of $\mathbb{Z}^2$ as \[(a,b)\times(c,d)=(ac+bd,ad+bc)\] for all integers $a,b,c,$ and $d$. Compute the number of ordered six-tuples $(a_1,a_2,a_3,a_4,a_5,a_6)$ of integers such that \[[[[[(1,a_1)\times (2,a_2)]\times (3,a_3)]\times (4,a_4)]\times (5,a_5)]\times (6,a_6)=(350,280).\]
[i]Proposed by Michael Ren and James Lin[/i]
2000 Harvard-MIT Mathematics Tournament, 40
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ and relatively prime to $n$. Find all natural numbers $n$ and primes $p$ such that $\phi(n)=\phi(np)$.
2000 Harvard-MIT Mathematics Tournament, 4
On an $n$ by $n$ chessboard, numbers are written on each square so that the number in a square is the average of the numbers on the adjacent squares. Show that all the numbers are the same.
1986 China National Olympiad, 6
Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.
2016 Korea USCM, 5
For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of
$$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$
2013 IFYM, Sozopol, 3
Determine all pairs $(p, q)$ of prime numbers such that $p^p + q^q + 1$ is divisible by $pq.$
2014 Sharygin Geometry Olympiad, 24
A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.
2024 Princeton University Math Competition, A6 / B8
Let $\triangle ABC$ be a triangle with $AB = 10.$ Let $D$ be a point on the opposite side of line $AC$ as $B$ so that $\triangle ACD$ is directly similar to $\triangle ABC$ (i.e. $\angle ACD = \angle ABC,$ etc). Let $M$ be the midpoint of $AD.$ Given that $A$ is the centroid of triangle $\triangle BCM,$ compute $BC^2.$
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2009 Tournament Of Towns, 3
Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ?
[i](4 points)[/i]
2018 India IMO Training Camp, 2
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
Russian TST 2018, P2
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
2014 Sharygin Geometry Olympiad, 17
Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.
2025 China Team Selection Test, 3
Let $n, k, l$ be positive integers satisfying $n \ge 3$, $l \le n - 2, l - k \le \frac{n-3}{2}$. Suppose that $a_1, a_2, \dots, a_k$ are integers chosen from $\{1, 2, \dots, n\}$ such that the set of remainders of the subset sums over all subsets of $a_i$ when divided by $n$ is exactly $\{1, 2, \dots, l\}$. Show that \[ a_1 + a_2 + \dots + a_k = l. \]
2025 Belarusian National Olympiad, 9.2
Snow White and seven dwarfs live in their house in the forest. During several days some dwarfs worked in the diamond mine, while others were collecting mushrooms. Each dwarf each day was doing only one type of job. It is known that in any two consecutive days there are exactly three dwarfs which did both types of job. Also, for any two days at least one dwarf did both types of job.
What is maximum amount of days which this situation could last?
[i]M. Karpuk[/i]
1990 AMC 8, 8
A dress originally priced at 80 dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was
$ \text{(A)}\ \text{45 dollars}\qquad\text{(B)}\ \text{52 dollars}\qquad\text{(C)}\ \text{54 dollars}\qquad\text{(D)}\ \text{66 dollars}\qquad\text{(E)}\ \text{68 dollars} $
1988 Tournament Of Towns, (168) 1
We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number.
(Folklore)
2004 Harvard-MIT Mathematics Tournament, 1
How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities?
\begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}
2018 AMC 10, 15
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[asy]
draw(circle((0,0),13));
draw(circle((5,-6.2),5));
draw(circle((-5,-6.2),5));
label("$B$", (9.5,-9.5), S);
label("$A$", (-9.5,-9.5), S);
[/asy]
$\textbf{(A) } 21 \qquad \textbf{(B) } 29 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 69 \qquad \textbf{(E) } 93 $
1966 All Russian Mathematical Olympiad, 072
There is exactly one astronomer on every planet of a certain system. He watches the closest planet. The number of the planets is odd and all of the distances are different. Prove that there one planet being not watched.
2003 Switzerland Team Selection Test, 8
Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$.
Suppose that there are circles $\omega_2,...,\omega_7$ such that:
(a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$,
(b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$.
Prove that $\omega_1 = \omega_7$.
2019 Saudi Arabia Pre-TST + Training Tests, 2.3
Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.
2002 India IMO Training Camp, 9
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.
If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?
2023 Turkey Team Selection Test, 4
Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$.
Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$
2018 Azerbaijan Senior NMO, 5
Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then
\[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\]
[i]A. Golovanov[/i]