Found problems: 85335
2021 Canadian Mathematical Olympiad Qualification, 6
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation
$$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$
2014 China Team Selection Test, 5
Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.
2013 Thailand Mathematical Olympiad, 12
Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$ at $Q$ (closer to $B$) and intersect $AM$ at $R$. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.
2016 IMO, 4
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
1970 IMO Longlists, 3
Prove that $(a!\cdot b!) | (a+b)!$ $\forall a,b\in\mathbb{N}$.
1998 VJIMC, Problem 4-I
Prove that there exists a program in standard Pascal which prints out its own ASCII code. No disk operations are permitted.
2001 Tournament Of Towns, 3
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.
1978 IMO Longlists, 6
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.
2021 Science ON grade X, 1
Consider the complex numbers $x,y,z$ such that
$|x|=|y|=|z|=1$. Define the number
$$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$
$\textbf{(a)}$ Prove that $a$ is a real number.
$\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$.
[i] (Stefan Bălăucă & Vlad Robu)[/i]
2015 Online Math Open Problems, 16
Given a (nondegenrate) triangle $ABC$ with positive integer angles (in degrees), construct squares $BCD_1D_2, ACE_1E_2$ outside the triangle. Given that $D_1, D_2, E_1, E_2$ all lie on a circle, how many ordered triples $(\angle A, \angle B, \angle C)$ are possible?
[i]Proposed by Yang Liu[/i]
2009 Regional Competition For Advanced Students, 2
How many integer solutions $ (x_0$, $ x_1$, $ x_2$, $ x_3$, $ x_4$, $ x_5$, $ x_6)$ does the equation
\[ 2x_0^2\plus{}x_1^2\plus{}x_2^2\plus{}x_3^2\plus{}x_4^2\plus{}x_5^2\plus{}x_6^2\equal{}9\]
have?
2013 Stars Of Mathematics, 1
Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have
\[m_a \leq f(a) + f^{-1}(a) \leq M_a.\]
[i](Dan Schwarz)[/i]
2007 Stanford Mathematics Tournament, 9
Peter Pan and Crocodile are each getting hired for a job. Peter wants to get paid 6.4 dollars daily, but Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, 80 cents on day 4, and so on. After how many whole days will Crocodile's total earnings exceed that of Peter's?
2020 Brazil EGMO TST, 1
Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.
1972 Spain Mathematical Olympiad, 3
Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.
2016 NZMOC Camp Selection Problems, 9
An $n$-tuple $(a_1, a_2 . . . , a_n)$ is [i]occasionally periodic[/i] if there exist a non-negative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+j+p}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, a_2 . . . , a_n)$ with elements from the set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, a_2, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
1987 IMO, 3
Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.
2024 Harvard-MIT Mathematics Tournament, 28
Given that the $32$-digit integer $$64 \ 312 \ 311 \ 692 \ 944 \ 269 \ 609 \ 355 \ 712 \ 372 \ 657$$ is the product of $6$ consecutive primes, compute the sum of these $6$ primes.
2024 Putnam, A4
Find all primes $p>5$ for which there exists an integer $a$ and an integer $r$ satisfying $1\leq r\leq p-1$ with the following property: the sequence $1,\,a,\,a^2,\,\ldots,\,a^{p-5}$ can be rearranged to form a sequence $b_0,\,b_1,\,b_2,\,\ldots,\,b_{p-5}$ such that $b_n-b_{n-1}-r$ is divisible by $p$ for $1\leq n\leq p-5$.
MathLinks Contest 4th, 6.3
If $n>2$ is an integer and $x_1, \ldots ,x_n$ are positive reals such that
\[ \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n \] then the following inequality takes place
\[ \frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}. \]
2017 AMC 12/AHSME, 6
The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point?
$\textbf{(A)}\ 4\sqrt2 \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt2 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6\sqrt2$
2009 Stanford Mathematics Tournament, 5
Find the minimum possible value of $2x^2+2xy+4y+5y^2-x$ for real numbers $x$ and $y$.
2025 Taiwan Mathematics Olympiad, 5
Two fixed circles $\omega$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $C$ and $D$ be two fixed points on the circle $\omega$. Let $P$ be a moving point on $\omega$. Line $PA$ meets circle $\Omega$ again at $Q$. Prove that the second intersection $R$ of two circumcircles of triangles $QPC$ and $QBD$ always lies on a fixed circle.
[i]Proposed by buratinogigle[/i]
2023 Durer Math Competition Finals, 15
Csongi bought a $12$-sided convex polygon-shaped pizza. The pizza has no interior point with three or more distinct diagonals passing through it. Áron wants to cut the pizza along $3$ diagonals so that exactly $6$ pieces of pizza are created. In how many ways can he do this? Two ways of slicing are different if one of them has a cut line that the other does not have.
2001 Argentina National Olympiad, 6
Given a rectangle $\mathcal{R}$ of area $100000 $, Pancho must completely cover the rectangle $\mathcal{R}$ with a finite number of rectangles with sides parallel to the sides of $\mathcal{R}$ . Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than $0.00001$, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,