Found problems: 85335
2014 Contests, 1
For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.)
[i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]
2014 Romania National Olympiad, 1
For a ring $ A, $ and an element $ a $ of it, define $ s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$
[b]a)[/b] Prove that if $ A $ is finite, then $ s_a $ is injective if and only if $ d_a $ is injective.
[b]b)[/b] Give example of a ring which has an element $ b $ for which $ s_b $ is injective and $ d_b $ is not, or, conversely, $ s_b $ is not injective, but $ d_b $ is.
1969 AMC 12/AHSME, 1
When $x$ is added to both the numerator and the denominator of the fraction $a/b, a\neq b, b\neq 0$, the value of the fraction is changed to $c/d$. Then $x$ equals:
$\textbf{(A) }\dfrac1{c-d}\qquad
\textbf{(B) }\dfrac{ad-bc}{c-d}\qquad
\textbf{(C) }\dfrac{ad-bc}{c+d}\qquad
\textbf{(D) }\dfrac{bc-ad}{c-d}\qquad
\textbf{(E) }\dfrac{bc-ad}{c+d}$
2008 Austria Beginners' Competition, 1
Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.
2022 Israel TST, 3
Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.
2007 Stars of Mathematics, 1
Prove that for every non-negative integer $ n, $ there exists a non-negative integer $ m $ such that
$$ \left( 1+\sqrt{2} \right)^n=\sqrt m +\sqrt{m+1} . $$
2005 Putnam, B3
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
2010 HMNT, 3
Triangle $ABC$ has $AB = 5$, $BC = 7$, and $CA = 8$. New lines not containing but parallel to $AB$, $BC$, and $CA$ are drawn tangent to the incircle of $ABC$. What is the area of the hexagon formed by the sides of the original triangle and the newly drawn lines?
1990 Austrian-Polish Competition, 7
$D_n$ is a set of domino pieces. For each pair of non-negative integers $(a, b)$ with $a \le b \le n$, there is one domino, denoted $[a, b]$ or $[b, a]$ in $D_n$. A [i]ring [/i] is a sequence of dominoes $[a_1, b_1], [a_2, b_2], ... , [a_k, b_k]$ such that $b_1 = a_2, b_2 = a_3, ... , b_{k-1} = a_k$ and $b_k = a_1$. Show that if $n$ is even there is a ring which uses all the pieces. Show that for n odd, at least $(n+1)/2$ pieces are not used in any ring. For $n$ odd, how many different sets of $(n+1)/2$ are there, such that the pieces not in the set can form a ring?
1980 IMO, 19
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
2011 Saudi Arabia Pre-TST, 1.2
Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.
2018 Tajikistan Team Selection Test, 8
Problem 8. For every non-negative integer n, define an n-variable function K_n (x_1,x_2,…,x_n ) as follows:
K_0=1
K_1 (x_1 )=〖x_1〗^2
K_(n+2) (x_1,x_2,…,x_(n+2) )=〖x_(n+2)〗^2.K_(n+1) (x_1,x_2,…,x_(n+1) )+(x_(n+2)+x_(n+1))K_n (x_1,x_2,…,x_n )
Prove that:
K_n (x_1,x_2,…,x_n )=K_n (x_n,…〖,x〗_2,x_1 )
2019 China Team Selection Test, 5
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
2025 Sharygin Geometry Olympiad, 12
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
2008 Bosnia and Herzegovina Junior BMO TST, 1
Let $ a,b,c$ be real positive numbers such that absolute difference between any two of them is less than $ 2$. Prove that: $ a \plus{} b \plus{} c < \sqrt {ab \plus{} 1} \plus{} \sqrt {ac \plus{} 1} \plus{} \sqrt {bc \plus{} 1}$
2022 Harvard-MIT Mathematics Tournament, 7
Find, with proof, all functions $f : R - \{0\} \to R$ such that $$f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))$$ for all real $x, y, z$ such that $xyz = 1$.
2022 Rioplatense Mathematical Olympiad, 1
Find three consecutive odd numbers $a,b,c$ such that $a^2+b^2+c^2$ is a four digit number with four equal digits.
2000 Harvard-MIT Mathematics Tournament, 22
Find the smallest $n$ such that $2^{2000}$ divides $n!$.
1965 Putnam, A6
In the plane with orthogonal Cartesian coordinates $x$ and $y$, prove that the line whose equation is $ux+vy = 1$ will be tangent to the cirve $x^m+y^m=1$ (where $m>1$) if and only if $u^n + v^n = 1$ and $m^{-1} + n^{-1} = 1$.
2009 China Team Selection Test, 2
In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$
2004 USAMO, 5
Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.
2008 China Team Selection Test, 1
Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.
2022 Girls in Mathematics Tournament, 4
The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?
2017 AMC 10, 21
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2010 Czech And Slovak Olympiad III A, 5
On the board are written numbers $1, 2,. . . , 33$. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least $16$ numbers remain on the board.