Found problems: 85335
2008 Danube Mathematical Competition, 1
$x,y,z,t \in \mathbb R_+^*$:
\[ (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} \]
1957 Poland - Second Round, 6
Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e. an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other.
2009 District Olympiad, 2
Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$
2012 Bulgaria National Olympiad, 3
We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:
\[*x^4+*x^3+*x^2+*x^1+*=0\]
with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy
1982 Canada National Olympiad, 2
If $a$, $b$ and $c$ are the roots of the equation $x^3 - x^2 - x - 1 = 0$,
(i) show that $a$, $b$ and $c$ are distinct:
(ii) show that
\[\frac{a^{1982} - b^{1982}}{a - b} + \frac{b^{1982} - c^{1982}}{b - c} + \frac{c^{1982} - a^{1982}}{c - a}\]
is an integer.
2007 AMC 12/AHSME, 17
If $ a$ is a nonzero integer and $ b$ is a positive number such that $ ab^{2} \equal{} \log_{10}b,$ what is the median of the set $ \{0,1,a,b,1/b\}$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ a \qquad \textbf{(D)}\ b \qquad \textbf{(E)}\ \frac {1}{b}$
2024 Malaysian Squad Selection Test, 7
Let $P$ be the set of all primes. Given any positive integer $n$, define $$\displaystyle f(n) = \max_{p \in P}v_p(n)$$ Prove that for any positive integer $k\ge 2$, there exists infinitely many positive integers $m$ such that \[ f(m+1) = f(m+2) = \cdots = f(m+k) \]
[i]Proposed by Ivan Chan Guan Yu[/i]
2016 Dutch IMO TST, 4
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
2011 AIME Problems, 12
Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.
1988 Mexico National Olympiad, 6
Consider two fixed points $B,C$ on a circle $w$. Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$.
2020 Bangladesh Mathematical Olympiad National, Problem 6
$f$ is a one-to-one function from the set of positive integers to itself such that $$f(xy) = f(x) × f(y)$$ Find the minimum possible value of $f(2020)$.
2011 VJIMC, Problem 4
Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.
2015 Iran MO (3rd round), 4
$p(x)\in \mathbb{C}[x]$ is a polynomial such that:
$\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R}$
Prove that $p(x)$ is constant.
1972 AMC 12/AHSME, 7
If $yz:zx:xy=1:2:3$, then $\dfrac{x}{yz}:\dfrac{y}{zx}$ is equal to
$\textbf{(A) }3:2\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:4\qquad\textbf{(D) }2:1\qquad \textbf{(E) }4:1$
1989 IMO Longlists, 65
Let $ ABCD$ be a quadrilateral inscribed in a circle of radius $ AB$ such that $ BC \equal{} a, CD \equal{} b,$ $ DA \equal{} \frac{3 \sqrt{3} \minus{} 1}{2} \cdot a$ For each point $ M$ on the semicircle with radius $ AB$ not containing $ C$ and $ D,$ denote by $ h_1, h_2, h_3$ the distances from $ M$ to the straight lines (sides) $ BC, CD,$ and $ DA.$ Find the maximum of $ h_1 \plus{} h_2 \plus{} h_3.$
2002 AIME Problems, 12
A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_{n}\le .4$ for all $n$ such that $1\le n \le 9$ is given to be $p^{a}q^{b}r/(s^{c}),$ where $p,$ $q,$ $r,$ and $s$ are primes, and $a,$ $b,$ and $c$ are positive integers. Find $(p+q+r+s)(a+b+c).$
1992 Putnam, B2
For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^{k}$ in the expansion $(1+x+x^{2}+x^{3})^{n}$. Prove that
$Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}$.
[hide="hint"]
Think of $\binom{n}{j}$ as the number of ways you can pick the $x^{2}$ term in the expansion.[/hide]
2009 Indonesia TST, 1
Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?
2019 IFYM, Sozopol, 3
The natural number $n>1$ is such that there exist $a\in \mathbb{N}$ and a prime number $q$ which satisfy the following conditions:
1) $q$ divides $n-1$ and $q>\sqrt{n}-1$
2) $n$ divides $a^{n-1}-1$
3) $gcd(a^\frac{n-1}{q}-1,n)=1$.
Is it possible for $n$ to be a composite number?
2024 Korea - Final Round, P4
For a triangle $ABC$, $O$ is the circumcircle and $D$ is a point on ray $BA$. $E$ and $F$ are points on $O$ so that $DE$ and $DF$ are tangent to $O$ and $EF$ cuts $AC$ at $T(\neq C)$. $P(\neq B,C)$ is a point on the arc $BC$ not containing $A$, and $DP$ cuts $O$ at $Q (\neq P)$. Let $BQ$ and $DT$ meets on $X (\neq Q)$, and $PT$ cuts $O$ at $Y (\neq P)$. Prove that $C,X,Y$ are collinear.
2000 Stanford Mathematics Tournament, 22
An equilateral triangle with sides of length $4$ has an isosceles triangle with the same base and half the height cut out of it.
Find the remaining area
1994 AMC 8, 6
The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is
$\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$
2006 China Team Selection Test, 2
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.
Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
2007 South East Mathematical Olympiad, 4
A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i=2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
2024 Romania National Olympiad, 2
Let $A \in \mathcal{M}_n(\mathbb{R})$ be an invertible matrix.
a) Prove that the eigenvalues of $AA^T$ are positive real numbers.
b) We assume that there are two distinct positive integers, $p$ and $q$, such that $(AA^T)^p=(A^TA)^q.$ Prove that $A^T=A^{-1}.$