Found problems: 85335
2003 Croatia Team Selection Test, 3
For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?
2007 Hanoi Open Mathematics Competitions, 4
List the numbers$\sqrt{2}; \sqrt[3]{3}; \sqrt[4]{4}; \sqrt[5]{5}; \sqrt[6]{6}.$ in order from greatest to least.
2008 Romania Team Selection Test, 2
Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$.
2004 Germany Team Selection Test, 3
Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
[i]Proposed by Hojoo Lee, Korea[/i]
1940 Moscow Mathematical Olympiad, 054
Factor $(b - c)^3 + (c - a)^3 + (a - b)^3$.
1994 ITAMO, 5
Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.
IV Soros Olympiad 1997 - 98 (Russia), 11.2
Find the three-digit number that has the greatest number of different divisors.
2013 Turkmenistan National Math Olympiad, 3
If a,b,c positive numbers and such that $a+\sqrt{b+\sqrt{c}}=c+\sqrt{b+\sqrt{a}}$. Prove that if $a\neq c$ then $40ac<1$.
STEMS 2021 Math Cat C, Q4
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
2012 Spain Mathematical Olympiad, 2
A sequence $(a_n)_{n\ge 1}$ of integers is defined by the recurrence
\[a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.\]
Prove that all terms of the sequence are integers and find an explicit formula for $a_n$.
2022 Pan-American Girls' Math Olympiad, 5
Find all positive integers $k$ for which there exist $a$, $b$, and $c$ positive integers such that \[\lvert (a-b)^3+(b-c)^3+(c-a)^3\rvert=3\cdot2^k.\]
2012 National Olympiad First Round, 11
The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$
2024 South Africa National Olympiad, 2
Determine which of the following is larger:
\[ \sqrt{2+\sqrt[3]{5}}\qquad \text{or}\qquad \sqrt[3]{5+\sqrt{2}}.\]
Fully explain your reasoning.
2011 IMO Shortlist, 4
For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences
\[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4.
[i]Proposed by Gerhard Wöginger, Austria[/i]
1991 All Soviet Union Mathematical Olympiad, 551
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
2011 Grand Duchy of Lithuania, 2
Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality
$(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$
2017 NIMO Problems, 5
In triangle $ABC$, $AB=12$, $BC=17$, and $AC=25$. Distinct points $M$ and $N$ lie on the circumcircle of $ABC$ such that $BM=CM$ and $BN=CN$. If $AM + AN = \tfrac{a\sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $\gcd(a, c) = 1$ and $b$ is not divisible by the square of a prime, compute $100a+10b+c$.
[i]Proposed by Michael Tang[/i]
2020 AMC 8 -, 21
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A [i]step[/i] consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)
[asy]//diagram by SirCalcsALot
size(200); int[] x = {6, 5, 4, 5, 6, 5, 6}; int[] y = {1, 2, 3, 4, 5, 6, 7}; int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy]
$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 35$
2001 Brazil Team Selection Test, Problem 3
For which positive integers $n$ is there a permutation $(x_1,x_2,\ldots,x_n)$ of $1,2,\ldots,n$ such that all the differences $|x_k-k|$, $k = 1,2,\ldots,n$, are distinct?
2010 Princeton University Math Competition, 2
Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
2009 Princeton University Math Competition, 5
Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.
2015 Saudi Arabia JBMO TST, 3
A natural number is called $nice$ if it doesn't contain 0 and if we add the product of its digit to the number, we obtain number with the same product of its digits. Prove that there is a nice 2015-digit number.
2015 Saudi Arabia GMO TST, 1
Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$
Trần Nam Dũng
2017 Morocco TST-, 4
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
2014 Contests, 1
Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.