Found problems: 85335
2010 International Zhautykov Olympiad, 1
Find all primes $p,q$ such that $p^3-q^7=p-q$.
2020 CMIMC Algebra & Number Theory, 1
Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.
MOAA Team Rounds, 2019.2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
2025 Euler Olympiad, Round 1, 7
Let $s(n)$ be the final value obtained after repeatedly summing the digits of $n$ until a single-digit number is reached. (For example: $s(187) = 7$, because the digit sum of $187$ is $16$ and the digit sum of $16$ is $7$). Evaluate the sum:
$$ s(1^2) + s(2^2) + s(3^2) + \ldots + s(2025^2)$$
[i]Proposed by Lia Chitishvili, Georgia [/i]
1999 Italy TST, 1
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.
1985 Spain Mathematical Olympiad, 3
Solve the equation $tan^2 2x+2 tan2x tan3x = 1$
2023 Greece Junior Math Olympiad, 4
Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.
2011 Princeton University Math Competition, A5 / B7
Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$ and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.
2005 MOP Homework, 2
Determine if there exist four polynomials such that the sum of any three of them has a real root while the sum of any two of them does not.
2021 AMC 12/AHSME Spring, 12
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
$\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$
1982 Bundeswettbewerb Mathematik, 2
In a convex quadrilateral $ABCD$ sides $AB$ and $DC$ are both divided into $m$ equal parts by points $A, S_1 , S_2 , \ldots , S_{m-1} ,B$ and $D,T_1, T_2, \ldots , T_{m-1},C,$ respectively (in this order).
Similarly, sides $BC$ and $AD$ are divided into $n$ equal parts by points $B,U_1,U_2, \ldots, U_{n-1},C$ and $A,V_1,V_2, \ldots,V_{n-1}, D$. Prove that for $1 \leq i \leq m-1$ each of the segments $S_i T_i$ is divided by the segments $U_j V_j$ ($1\leq j \leq n-1$) into $n$ equal parts
2006 Mathematics for Its Sake, 2
The cevians $ AP,BQ,CR $ of the triangle $ ABC $ are concurrent at $ F. $ Prove that the following affirmations are equivalent.
$ \text{(i)} \overrightarrow{AP} +\overrightarrow{BQ} +\overrightarrow{CR} =0 $
$ \text{(ii)} F$ is the centroid of $ ABC $
[i]Doru Isac[/i]
1992 Hungary-Israel Binational, 5
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$
\[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \]
where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Show that $L_{2n+1}+(-1)^{n+1}(n \geq 1)$ can be written as a product of three (not necessarily distinct) Fibonacci numbers.
2004 Croatia National Olympiad, Problem 1
Parts of a pentagon have areas $x,y,z$ as shown in the picture. Given the area $x$, find the areas $y$ and $z$ and the area of the entire pentagon.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS9mLzM5NjNjNDcwY2ZmMzgzY2QwYWM0YzI1NmYzOWU2MWY1NTczZmYxLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wOCBhdCA0LjMwLjU1IFBNLnBuZw[/img]
2018 Saudi Arabia GMO TST, 2
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
2014 Canada National Olympiad, 4
The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.
2001 Mongolian Mathematical Olympiad, Problem 1
Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies
$$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.
India EGMO 2025 TST, 3
Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that
$$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$
Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$.
[i] Proposed by Antareep Nath [/i]
2024 Harvard-MIT Mathematics Tournament, 10
A [i]peacock [/i] is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.
1990 French Mathematical Olympiad, Problem 3
(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$.
(b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.
2012 QEDMO 11th, 1
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
2020 JBMO TST of France, 3
Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive
if, for any real numbers $x_1, x_2,...,x_n$
with the property that $x_1+x_2+...+x_n = 0$,
the inequality $f(x_1)+f(x_2)+...+f(x_n)=>0$ is true
a) Is it true that any 2020-positive function is also 1010-positive?
b) Is it true that any 1010-positive function is 2020-positive?
1977 AMC 12/AHSME, 28
Let $g(x)=x^5+x^4+x^3+x^2+x+1$. What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$?
$\textbf{(A) }6\qquad\textbf{(B) }5-x\qquad\textbf{(C) }4-x+x^2\qquad$
$\textbf{(D) }3-x+x^2-x^3\qquad \textbf{(E) }2-x+x^2-x^3+x^4$
2021 Junior Balkan Team Selection Tests - Moldova, 1
Find all values of the real parameter $a$, for which the equation $(x -6\sqrt{x} + 8)\cdot \sqrt{x- a} = 0$ has exactly two distinct real solutions.
1998 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.