Found problems: 85335
1989 Greece Junior Math Olympiad, 1
Let $A$ be the sum of three consecutive integers and $B$ be the sum of the exact three consecutive integers. Is it possible to have $AB=33333$ ?
2014 BMT Spring, 19
Evaluate the integral $\int_0^{\pi/2} \sqrt{\tan \theta} d\theta$.
2006 Princeton University Math Competition, 3
Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.
2005 ITAMO, 3
Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.
2015 Balkan MO Shortlist, N1
Let $d$ be an even positive integer.
John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$
He continues until two numbers remain written on on the blackboard.
Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$.
(Albania)
1998 Iran MO (3rd Round), 1
Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$,
[b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$,
[b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$,
[b](iii)[/b] $f(m)=m$ if and only if $m=1$.
2015 China Second Round Olympiad, 2
In isoceles $\triangle ABC$, $AB=AC$, $I$ is its incenter, $D$ is a point inside $\triangle ABC$ such that $I,B,C,D$ are concyclic. The line through $C$ parallel to $BD$ meets $AD$ at $E$. Prove that $CD^2=BD\cdot CE$.
2013 Cuba MO, 5
Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.
1996 Portugal MO, 1
Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.
2007 Vietnam National Olympiad, 3
Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A
2020 Taiwan TST Round 2, 4
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
[i]Proposed by YaWNeeT[/i]
2013 Puerto Rico Team Selection Test, 2
How many 3-digit numbers have the property that the sum of their digits is even?
1999 AMC 12/AHSME, 12
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
2006 Bulgaria Team Selection Test, 1
[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \]
[i] Stoyan Atanasov[/i]
1948 Moscow Mathematical Olympiad, 146
Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.
2008 AMC 10, 9
A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \frac{b}{a} \qquad
\textbf{(D)}\ \frac{2b}{a} \qquad
\textbf{(E)}\ \sqrt{2b\minus{}a}$
2024 HMNT, 3
Points $K,A,L,C,I,T,E$ are such that triangles $CAT$ and $ELK$ are equilateral, share a center $I,$ and points $E,L,K$ lie on sides $CA, AT, TC$ respectively. If the area of triangle $CAT$ is double the area of triangle $ELK$ and $CI = 2,$ compute the minimum possible value of $CK.$
2005 iTest, 27
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)
2017 Macedonia National Olympiad, Problem 1
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds:
$$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$
PEN H Problems, 33
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
2024 AMC 10, 8
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1{:}00 \text{ PM}$ and were able to pack $4$, $3$, and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45 \text{ PM}$. At what time did Daria join the group?
$\textbf{(A) }1{:}25\text{ PM}\qquad\textbf{(B) }1{:}35\text{ PM}\qquad\textbf{(C) }1{:}45\text{ PM}\qquad\textbf{(D) }1{:}55\text{ PM}\qquad\textbf{(E) }2{:}05\text{ PM}\qquad$
2013 Poland - Second Round, 5
Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.
2019 Romania Team Selection Test, 2
Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $
[i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]
2020 Online Math Open Problems, 11
A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
[i]Proposed by Sean Li[/i]
2018 Tajikistan Team Selection Test, 1
Problem 1. Let ω be the incircle of triangle ABC which is tangent to BC,CA,AB at points D,E,F, respectively. The altitudes of triangle DEF with respect to E,F meet AB,AC at points X,Y, respectively. Prove that the second intersection of the circumcircles of triangles AEX,AFY lies on the circle ω.