Found problems: 85335
2012 Today's Calculation Of Integral, 770
Find the value of $a$ such that :
\[101a=6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]
2007 Macedonia National Olympiad, 4
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy
\[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2})
\]
for all $ x, y \in\mathbb R.$
2007 Princeton University Math Competition, 3
In how many ways can $1 + 2 + \cdots + 2007$ be expressed as a sum of consecutive positive integers?
2007 iTest Tournament of Champions, 3
Find the largest natural number $n$ such that \[2^n + 2^{11} + 2^8\] is a perfect square.
2023 LMT Fall, 11
Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.
2010 IFYM, Sozopol, 2
Known $f:\mathbb{N}_0 \to \mathbb{N}_0$ function for $\forall x,y\in \mathbb{N}_0$ the following terms are paid
$(a). f(0,y)=y+1$
$(b). f(x+1,0)=f(x,1)$
$(c). f(x+1,y+1)=f(x,f(x+1,y)).$
Find the value if $f(4,1981)$
1995 AMC 8, 7
At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home?
$\text{(A)}\ \dfrac{1}{16} \qquad \text{(B)}\ \dfrac{3}{20} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{17}{20} \qquad \text{(E)}\ \dfrac{9}{10}$
2007 Moldova Team Selection Test, 1
Find the least positive integers $m,k$ such that
a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube.
b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square.
The author is Vasile Suceveanu
2024 Azerbaijan IMO TST, 6
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.
Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.
[i]Ivan Chan Kai Chin, Malaysia[/i]
2014 Singapore Junior Math Olympiad, 2
Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.
2023 All-Russian Olympiad, 7
Given a trapezoid $ABCD$, in which $AD \parallel BC$, and rays $AB$ and $DC$ intersect at point $G$. The common external tangents to the circles $(ABC), (ACD)$ intersect at point $E$. The common external tangents to circles $(ABD), (CBD)$ meet at $F$. Prove that the points $E, F$ and $G$ are collinear.
2007 China Team Selection Test, 2
Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to
$ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.
Geometry Mathley 2011-12, 2.4
Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$.
Nguyễn Minh Hà
2016 Saudi Arabia Pre-TST, 1.4
Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$.
1. Show that $F(r)$ is a positive integer for any prime $r \ne p$.
2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$.
3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.
2021 MIG, 11
The figure below is used to fold into a pyramid, and consists of four equilateral triangles erected around a square with area nine. What is the length of the dashed path shown?
[asy]
real r = 1/2 * 3^(1/2);
size(45);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((0,0)--(-r,0.5)--(0,1)--(0.5,1+r)--(1,1)--(1+r,0.5)--(1,0)--(0.5,-r)--cycle,dashed);
[/asy]
$\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }24\qquad\textbf{(E) }27$
2006 Tournament of Towns, 2
Suppose $ABC$ is an acute triangle. Points $A_1, B_1$ and $C_1$ are chosen on sides $BC, AC$ and $AB$ respectively so that the rays $A_1A, B_1B$ and $C_1C$ are bisectors of triangle $A_1B_1C_1$. Prove that $AA_1, BB_1$ and $CC_1$ are altitudes of triangle $ABC$. (6)
2022 Turkey EGMO TST, 3
Find all pairs of integers $(a,b)$ satisfying the equation $a^7(a-1)=19b(19b+2)$.
2007 Princeton University Math Competition, 3
Find all values of $b$ such that the difference between the maximum and minimum values of $f(x) = x^2-2bx-1$ on the interval $[0, 1]$ is $1$.
2008 Princeton University Math Competition, 9
Alex Lishkov is trying to guess sequence of $2009$ random ternary digits ($0, 1$, or $2$). After he guesses each digit, he finds out whether he was right or not. If he guesses incorrectly, and $k$ was the correct answer, then an oracle tells him what the next $k$ digits will be. Being Bulgarian, Lishkov plays to maximize the expected number of digits guessed correctly. Let $P_n$ be the probability that Lishkov guesses the nth digit correctly. Find $P_{2009}$. Write your answer in the form $x + yRe(\rho^k)$, where $x$ and $y$ are rational, $\rho$ is complex, and $k$ is a positive integer
1966 AMC 12/AHSME, 37
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals:
$\text{(A)}\ \dfrac{5}{2}\qquad
\text{(B)}\ \frac{3}{2}\qquad
\text{(C)}\ \dfrac{4}{3}\qquad
\text{(D)}\ \dfrac{5}{4}\qquad
\text{(E)}\ \dfrac{3}{4}$
1973 Czech and Slovak Olympiad III A, 6
Consider a square of side of length 50. A polygonal chain $L$ is given in the square such that for every point $P$ of the square there is a point $Q$ of the chain with the property $PQ\le 1.$ Show that the length of $L$ is greater than 1248.
2007 Hanoi Open Mathematics Competitions, 10
Let a; b; c be positive real numbers such that $\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab} \geq 1$. Prove that $\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \geq 1$.
2023 Durer Math Competition Finals, 6
In Eldorado a year has $20$ months, and each month has $20$ days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the $19$th day of the $4$th month, she would say $4 \cdot 19 = 76$.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?
2018 ISI Entrance Examination, 7
Let $a, b, c$ are natural numbers such that $a^{2}+b^{2}=c^{2}$ and $c-b=1$
Prove that
$(i)$ $a$ is odd.
$(ii)$ $b$ is divisible by $4$
$(iii)$ $a^{b}+b^{a}$ is divisible by $c$
2023 Azerbaijan National Mathematical Olympiad, 2
Let $I$ be the incenter in the acute triangle $ABC.$ Rays $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ at points $S$ and $T,$ respectively. The segment $ST$ intersects the sides $AB$ and $AC$ at points $K$ and $L,$ respectively. Prove that $AKIL$ is a rhombus.