Found problems: 85335
1990 Hungary-Israel Binational, 1
Prove that there are no positive integers $x$ and $y$ such that $x^2+y+2$ and $y^2+4x$ are perfect squares
I Soros Olympiad 1994-95 (Rus + Ukr), 9.2
Triangles $MA_2B_2$ and $MA_1B_1$ are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines $A_1A_2$ , $B_1B_2$ have a common point.
2023 Romania Team Selection Test, P3
Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$
[i]Proposed by Andrei Bâra[/i]
2009 Kosovo National Mathematical Olympiad, 4
$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that
$(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$.
$(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.
1990 Tournament Of Towns, (264) 2
The vertices of an equilateral triangle lie on sides $ AB$, $CD$ and $EF$ of a regular hexagon $ABCDEF$. Prove that the triangle and the hexagon have a common centre.
(N Sedrakyan, Yerevan )
2021 Saudi Arabia Training Tests, 17
Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Tangent lines to $(O)$ at $B,C$ meet at $T$. A line passes through $T$ cuts segments $AB$ at $D$ and cuts ray $CA$ at $E$. Take $M$ as midpoint of $DE$ and suppose that $MA$ cuts $(O)$ again at $K$. Prove that $(MKT)$ is tangent to $(O)$.
2016 CCA Math Bonanza, T5
How many permutations of the word ``ACADEMY'' have that there exist two vowels that are separated by an odd distance? For example, the X and Y in XAY are separated by an even distance, while the X and Y in XABY are separated by an odd distance. Note: the vowels are A, E, I, O, and U. Y is [b]NOT[/b] a vowel.
[i]2016 CCA Math Bonanza Team #5[/i]
2007 Thailand Mathematical Olympiad, 13
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
2021 Cono Sur Olympiad, 2
Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
MBMT Team Rounds, 2020.4
Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number.
[i]Proposed by Gabriel Wu[/i]
2020 HMNT (HMMO), 3
Harvard has recently built a new house for its students consisting of $n$ levels, where the $k$th level from the top can be modeled as a $1$-meter-tall cylinder with radius $k$ meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is $35$ percent of the total surface area of the building (including the bottom), compute $n$.
1985 AMC 12/AHSME, 9
The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Counting from the left, the column in which $ 1985$ appears in is the
[asy]
int i,j;
for(i=0; i<4; i=i+1) {
label(string(16*i+1), (2*1,-2*i));
label(string(16*i+3), (2*2,-2*i));
label(string(16*i+5), (2*3,-2*i));
label(string(16*i+7), (2*4,-2*i));
}
for(i=0; i<3; i=i+1) {
for(j=0; j<4; j=j+1) {
label(string(16*i+15-2*j), (2*j,-2*i-1));
}}
dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10));
for(i=-10; i<-6; i=i+1) {
for(j=1; j<4; j=j+1) {
dot((2*j,i));
}}
[/asy]
$ \textbf{(A)} \text{ first} \qquad \textbf{(B)} \text{ second} \qquad \textbf{(C)} \text{ third} \qquad \textbf{(D)} \text{ fourth} \qquad \textbf{(E)} \text{ fifth}$
2007 Pre-Preparation Course Examination, 1
$D$ is an arbitrary point inside triangle $ABC$, and $E$ is inside triangle $BDC$. Prove that \[\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}\]
2020 LMT Fall, A19
Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying the $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?
[i]Proposed by Janabel Xia[/i]
2014 Romania National Olympiad, 4
Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.
2011 Croatia Team Selection Test, 3
Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.
Kyiv City MO Juniors 2003+ geometry, 2017.8.4
On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $N$ are selected in such a way that $\angle MAN= 45^o$. Using the segment $MN$, as the diameter, we constructed a circle $w$, which intersects the segments $AM$ and $AN$ at points $P$ and $Q$, respectively. Prove that the points $B, P$ and $Q$ lie on the same line.
1970 Canada National Olympiad, 10
Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.
2021 Girls in Mathematics Tournament, 2
Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?
1991 Bulgaria National Olympiad, Problem 3
Prove that for every prime number $p\ge5$,
(a) $p^3$ divides $\binom{2p}p-2$;
(b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.
1985 IMO Longlists, 26
Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that
\[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]
2019 Romania National Olympiad, 2
Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$
Prove that $f$ is constant.
1986 Polish MO Finals, 3
$p$ is a prime and $m$ is a non-negative integer $< p-1$.
Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.
2022 CMIMC, 9
For natural numbers $n$, let $r(n)$ be the number formed by reversing the digits of $n$, and take $f(n)$ to be the maximum value of $\frac{r(k)}k$ across all $n$-digit positive integers $k$.
If we define $g(n)=\left\lfloor\frac1{10-f(n)}\right\rfloor$, what is the value of $g(20)$?
[i]Proposed by Adam Bertelli[/i]
2016 Iran Team Selection Test, 3
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.