Found problems: 85335
2025 Bangladesh Mathematical Olympiad, P4
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$
[i]Proposed by Md. Fuad Al Alam[/i]
2020 BMT Fall, 9
A sequence $a_n$ is defined by $a_0 = 0$, and for all $n \ge 1$, $a_n = a_{n-1} + (-1)^n \cdot n^2$. Compute $a_{100}$
2009 Indonesia TST, 2
Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).
2016 Oral Moscow Geometry Olympiad, 6
Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.
2016 India National Olympiad, P4
Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.
2021 JHMT HS, 3
Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$
2008 IberoAmerican, 4
Prove that the equation \[ x^{2008}\plus{} 2008!\equal{} 21^{y}\] doesn't have solutions in integers.
1999 Irish Math Olympiad, 1
Solve the system of equations:
$ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$
$ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$
2002 AMC 10, 4
For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many
2000 Brazil Team Selection Test, Problem 1
Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$.
[color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]
1999 Tournament Of Towns, 3
Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards.
(H. Lebesgue - A Kanel)
2015 Rioplatense Mathematical Olympiad, Level 3, 1
Let $ABC$ be a triangle and $P$ a point on the side $BC$. Let $S_1$ be the circumference with center $B$ and radius $BP$ that cuts the side $AB$ at $D$ such that $D$ lies between $A$ and $B$. Let $S_2$ be the circumference with center $C$ and radius $CP$ that cuts the side $AC$ at $E$ such that $E$ lies between $A$ and $C$. Line $AP$ cuts $S_1$ and $S_2$ at $X$ and $Y$ different from $P$, respectively. We call $T$ the point of intersection of $DX$ and $EY$. Prove that $\angle BAC+ 2 \angle DTE=180$
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
$$\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$