Found problems: 85335
2016 Online Math Open Problems, 5
Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$?
[i] Proposed by James Lin [/i]
2000 Bosnia and Herzegovina Team Selection Test, 1
Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$
1957 Miklós Schweitzer, 6
[b]6.[/b] Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form
$\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$).
From the coefficients $a_{kn}$ compose the sequence of polynomials
$P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ($n=0,1,2,\dots$)
and find a closed form for the function
$F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$
[b](S. 22)[/b]
India EGMO 2025 TST, 8
Let $ABCD$ be a trapezium with $AD||BC$; and let $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $\angle DAY=\angle CAB$ then the internal bisectors of $\angle XAY$ and $\angle XBY$ meet on $XY$.
Proposed by Belur Jana Venkatachala
2023 IMO, 1
Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.
1975 Bundeswettbewerb Mathematik, 3
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
2006 Junior Tuymaada Olympiad, 7
The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.
2010 LMT, 14
Al and Bob are joined by Carl and D’Angelo, and they decide to play a team game of Rock Paper Scissors. A game is called [i]perfect[/i] if some two of the four play the same thing, and the other two also play the same thing, but something different. For example, an example of a perfect game would be Al and Bob playing rock, and Carl and D’Angelo playing scissors, but if all four play paper, we do not have a perfect game. What is the probability of a perfect game?
2015 Thailand TSTST, 1
Let $A$ and $B$ be nonempty sets and let $f : A \to B$. Prove that the following statements are equivalent:
$\text{(i) }$ $f$ is surjective.
$\text{(ii)} $ For every set $C$ and and every functions $g, h : B \to C$, if $g\circ f = h \circ f$ then $g = h$.
2018 Hong Kong TST, 4
In triangle $ABC$ with incentre $I$, let $M_A,M_B$ and $M_C$ by the midpoints of $BC, CA$ and $AB$ respectively, and $H_A,H_B$ and $H_C$ be the feet of the altitudes from $A,B$ and $C$ to the respective sides. Denote by $\ell_b$ the line being tangent tot he circumcircle of triangle $ABC$ and passing through $B$, and denote by $\ell_b'$ the reflection of $\ell_b$ in $BI$. Let $P_B$ by the intersection of $M_AM_C$ and $\ell_b$, and let $Q_B$ be the intersection of $H_AH_C$ and $\ell_b'$. Defined $\ell_c,\ell_c',P_C,Q_C$ analogously. If $R$ is the intersection of $P_BQ_B$ and $P_CQ_C$, prove that $RB=RC$.
2011 Pre-Preparation Course Examination, 3
Calculate number of the hamiltonian cycles of the graph below: (15 points)
2014 Singapore Junior Math Olympiad, 3
In the triangle $ABC$, the bisector of $\angle A$ intersects the bisection of $\angle B$ at the point $I, D$ is the foot of the perpendicular from $I$ onto $BC$. Prove that the bisector of $\angle BIC$ is perpendicular to the bisector $\angle AID$.
2005 Austrian-Polish Competition, 3
Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions:
a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$.
b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$.
Prove that
$$\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$
1993 India National Olympiad, 3
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]
2014 Contests, 1
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
2008 Greece Team Selection Test, 1
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$
is divisible by $\phi (x)=x^2-x+1$
2014 BAMO, 3
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$.
2006 Mathematics for Its Sake, 1
[b]a)[/b] Show that there are $ 4 $ equidistant parallel planes that passes through the vertices of the same tetrahedron.
[b]b)[/b] How many such $ \text{4-tuplets} $ of planes does exist, in function of the tetrahedron?
2010 Purple Comet Problems, 16
Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?
1998 Tournament Of Towns, 5
The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?
(A Shen)
2012 Pre - Vietnam Mathematical Olympiad, 1
Let $n \geq 2$ be a positive integer. Suppose there exist non-negative integers ${n_1},{n_2},\ldots,{n_k}$ such that $2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}$. Prove that $k \ge n$.
2018 Estonia Team Selection Test, 12
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
2000 All-Russian Olympiad Regional Round, 9.5
In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$.
[img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]
2011 Tournament of Towns, 4
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?
2014 NIMO Problems, 5
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]