Found problems: 85335
2005 MOP Homework, 1
Find all triples $(x,y,z)$ such that $x^2+y^2+z^2=2^{2004}$.
2012 ELMO Problems, 5
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
2015 CCA Math Bonanza, T4
Evaluate the continued fraction $$1+\frac{2}{2+\frac{2}{2+\ldots}}$$
[i]2015 CCA Math Bonanza Team Round #4[/i]
1984 Putnam, A3
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by
$m_{ij}=x$ if $i=j$,
$m_{ij}=a$ if $i \not= j$ and $i+j$ is even,
$m_{ij}=b$ if $i \not= j$ and $i+j$ is odd.
For example
$ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a
& b& x & b\\ b & a & b & x \end{vmatrix}$.
Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ .
P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)
2014 Canadian Mathematical Olympiad Qualification, 3
Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 | (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.
2022-2023 OMMC, 12
Initially five variables are defined: $a_1=1, a_2=0, a_3=0, a_4=0, a_5=0.$ On a turn, Evan can choose an integer $2 \le i \le 5.$ Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2,$ then now $a_1=1, a_2=0+1=1, a_3=0, a_4=0, a_5=0.$ Find the minimum number of turns Evan needs to make $a_5$ exceed $1,000,000.$
1974 Spain Mathematical Olympiad, 2
In a metallic disk, a circular sector is removed, so that with the remaining can form a conical glass of maximum volume. Calculate, in radians, the angle of the sector that is removed.
[hide=original wording]En un disco metalico se quita un sector circular, de modo que con la parte restante se pueda formar un vaso c´onico de volumen maximo. Calcular, en radianes, el angulo del sector que se quita.[/hide]
2005 Iran MO (3rd Round), 2
We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i,
A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$.
Say the the following sets have the relation $\sim$ or not ?
a) Natural numbers and composite numbers.
b) Rational numbers and rational numbers with finite digits in base 10.
c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$
d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$
2013 HMIC, 2
Find all functions $f : R \to R$ such that, for all real numbers $x, y,$
$$(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).$$
2013 Albania Team Selection Test, 1
Find the 3-digit number whose ratio with the sum of its digits it's minimal.
1898 Eotvos Mathematical Competition, 1
Determine all positive integers $n$ for which $2^n + 1$ is divisible by $3$.
2016 Costa Rica - Final Round, N2
Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.
2007 Romania National Olympiad, 1
Prove that the number $ 10^{10}$ can't be written as the product of two natural numbers which do not contain the digit "$ 0$" in their decimal representation.
1996 All-Russian Olympiad Regional Round, 8.6
Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.
2011 Today's Calculation Of Integral, 752
Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$
2002 Vietnam National Olympiad, 2
Determine for which $ n$ positive integer the equation: $ a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd}$ has positive integer solutions.
2010 Indonesia TST, 4
How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $
2003 Switzerland Team Selection Test, 3
Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]
1986 Canada National Olympiad, 2
A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers. The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?
2000 Saint Petersburg Mathematical Olympiad, 9.4
On a Cartesian plane 101 planes are drawn and all points of intersection are labeled. Is it possible, that for every line, 50 of the points have positive coordinates and 50 of the points have negative coordinates
[I]Proposed by S. Ivanov[/i]
KoMaL A Problems 2023/2024, A. 877
A convex quadrilateral $ABCD$ is circumscribed about circle $\omega$. A tangent to $\omega$ parallel to $AC$ intersects $BD$ at a point $P$ outside of $\omega$. The second tangent from $P$ to $\omega$ touches $\omega$ at a point $T$. Prove that $\omega$ and circumcircle of $ATC$ are tangent.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
2011 All-Russian Olympiad, 4
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
[i]M. Kungojin[/i]
1979 AMC 12/AHSME, 18
To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$?
$\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$
2006 Korea Junior Math Olympiad, 4
In the coordinate plane, de fine $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is de fined on $M$, sends $(a,b)$
to $(a + b, b)$. Transformation $T$, also de fined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we
can use $S,T$ denitely to map it to $(g,0)$.
2023 AMC 12/AHSME, 2
The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza?
$\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$