Found problems: 85335
2023 Estonia Team Selection Test, 1
Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.
PEN G Problems, 29
Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?
2015 AMC 12/AHSME, 24
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number?
$\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$
2021 VIASM Math Olympiad Test, Problem 4
The number selection game is the following single-player game. Originally, on the table there were positive integers $1, 2,...,22$ (All positive integers not exceeding $22$ appear exactly once). In each move, the player chooses the three numbers $a, b, c$ that are on the table, then the selected numbers $a, b, c$ disappear but a new number $a + b + c$ appears; At the same time, the player's score is added $(a + b)(b+c)(c + a)$. The initial score was $0$. The game ends after $10$ moves (when there are only two numbers left on the board). Call $M, m$ respectively the highest and the lowest possible score of a game.
Determine the value of $\dfrac{M}{m}$.
2006 Taiwan National Olympiad, 3
Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.
2016 CHMMC (Fall), 12
For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.
1988 Irish Math Olympiad, 11
If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $1/a$, $a>0$, to use the iteration $$x_{k+1}=x_k(2-ax_k), \quad \quad k=0,1,2,\dots ,$$ where $x_0$ is a selected “starting” value. Find the limitations, if any, on the starting values $x_0$, in order that the above iteration converges to the desired value $1/a$.
2006 MOP Homework, 1
Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.
2012 Moldova Team Selection Test, 1
Prove that polynomial $x^8+98x^4+1$ can be factorized in $Z[X]$.
2007 Thailand Mathematical Olympiad, 18
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
2001 All-Russian Olympiad Regional Round, 9.4
The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?
2010 Junior Balkan Team Selection Tests - Romania, 2
Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$.
Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.
LMT Team Rounds 2010-20, B25
Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?
2018 Vietnam National Olympiad, 4
On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1,\, x_2,\, x_3$.
a. Prove that the following quantity is a constant:
$$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$
b. Prove the following inequality:
$$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$
2006 Turkey Team Selection Test, 3
Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list.
a) Prove that we can always find a good list with 12 elements, whatever the lists are;
b) Prove that students can make lists such that no shorter list is good.
2000 Austrian-Polish Competition, 6
Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.
1991 Romania Team Selection Test, 1
Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$, is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4.
[i]Laurentiu Panaitopol[/i]
2011 Canadian Mathematical Olympiad Qualification Repechage, 3
Determine all solutions to the system of equations:
\[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\]
[This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]
2011 ELMO Shortlist, 5
Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that
\[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$.
[i]Victor Wang.[/i]
2023 Philippine MO, 5
Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.
Ukraine Correspondence MO - geometry, 2007.9
In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.
2005 China Team Selection Test, 1
Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.
2023 Iran Team Selection Test, 6
Suppose that we have $2n$ non-empty subset of $ \big\{0,1,2,...,2n-1\big\} $ that sum of the elements of these subsets is $ \binom{2n+1}{2}$ . Prove that we can choose one element from every subset that some of them is $ \binom{2n}{2}$
[i]Proposed by Morteza Saghafian and Afrouz Jabalameli [/i]
2001 Regional Competition For Advanced Students, 4
Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of $n$.
2016 Hong Kong TST, 1
Find all natural numbers $n$ such that $n$, $n^2+10$, $n^2-2$, $n^3+6$, and $n^5+36$ are all prime numbers.