Found problems: 85335
2014 China Team Selection Test, 1
$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.
1990 Bulgaria National Olympiad, Problem 2
Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).
2006 AMC 8, 2
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 26$
2021 DIME, 14
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $$\sum_{n=1}^{210} nf(n)$$ is divided by $211$.
[i]Proposed by ApraTrip[/i]
2016 Latvia Baltic Way TST, 6
Given a natural number $n$, for which we can find a prime number less than $\sqrt{n}$ that is not a divisor of $n$. The sequence $a_1, a_2,... ,a_n$ is the numbers $1, 2,... ,n$ arranged in some order. For this sequence, we will find the longest ascending subsequense $a_{i_1} < a_{i_2} < ... < a_{i_k}$, ($i_1 <...< i_k$) and the longest decreasing substring $a_{j_1} > ... > a_{j_l}$, ($j_1 < ... < j_l$) . Prove that at least one of these two subsequnsces $a_{i_1} , . . . , a_{i_k}$ and $a_{j_1} > ... > a_{j_l}$ contains a number that is not a divisor of $n$.
2020 Colombia National Olympiad, 3
A number is said to be [i]triangular [/i] if it can be expressed in the form $1 + 2 +...+n$ for some positive integer $n$. We call a positive integer $a$ [i]retriangular [/i] if there exists a fixed positive integer $ b$ such that $aT +b$ is a triangular number whenever $T$ is a triangular number. Determine all retriangular numbers.
2002 Junior Balkan Team Selection Tests - Moldova, 2
$64$ distinct points are positioned in the plane so that they determine exactly $2003$ different lines. Prove that among the $64$ points there are at least $4$ collinear points.
2019 OMMock - Mexico National Olympiad Mock Exam, 4
Find all positive integers $n$ such that it is possible to split the numbers from $1$ to $2n$ in two groups $(a_1,a_2,..,a_n)$, $(b_1,b_2,...,b_n)$ in such a way that $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$.
[i]Proposed by Alef Pineda[/i]
2021 IMO Shortlist, G7
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.
2015 AMC 12/AHSME, 22
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
$ \textbf{(A) }14\qquad\textbf{(B) }16\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }24 $
2022 Junior Balkan Team Selection Tests - Moldova, 11
Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.
2019 District Olympiad, 1
Let $n \in \mathbb{N}, n \ge 2$ and the positive real numbers $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ such that $a_1+a_2+…+a_n=b_1+b_2+…+b_n=S.$
$\textbf{a)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}.$
$\textbf{b)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.$
2021 VIASM Math Olympiad Test, Problem 2
Given a square $5$ x $7$ board and $35$ pieces, each piece is formed by $3$ squares like below:
[size=75][center][img]https://i.ibb.co/hFDhp9p/Screenshot-2023-03-26-061057.png[/img][/center][/size]
Can we fill the board with $35$ pieces such that there are exactly $3$ pieces superimpose on every square of the given board?
[i][color=#E06666]Note: we can rotate, turn upside down the pieces[/color][/i]
2002 Balkan MO, 1
Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ($n\geq 4$) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$, such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$, and $X_{2k}$ is connected with $X_1$.
2018 Dutch IMO TST, 1
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
2003 Czech And Slovak Olympiad III A, 5
Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.
Today's calculation of integrals, 892
Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$
2020 USA IMO Team Selection Test, 2
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.
Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.
[i]Merlijn Staps[/i]
2020 Dutch BxMO TST, 5
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
2023 Silk Road, 2
Let $n$ be a positive integer. Each cell of a $2n\times 2n$ square is painted in one of the $4n^2$ colors (with some colors may be missing). We will call any two-cell rectangle a [i]domino[/I], and a domino is called [i]colorful[/I] if its cells have different colors. Let $k$ be the total number of colorful dominoes in our square; $l$ be the maximum integer such that every partition of the square into dominoes contains at least $l$ colorful dominoes. Determine the maximum possible value of $4l-k$ over all possible colourings of the square.
2022 239 Open Mathematical Olympiad, 2
Point $I{}$ is the center of the circle inscribed in the quadrilateral $ABCD$. Prove that there is a point $K{}$ on the ray $CI$ such that $\angle KBI=\angle KDI=\angle BAI$.
2022 Bolivia Cono Sur TST, P4
Find all right triangles with integer sides and inradius 6.
2018 Baltic Way, 18
Let $n \ge 3$ be an integer such that $4n+1$ is a prime number. Prove that $4n+1$ divides $n^{2n}-1$.
2009 Spain Mathematical Olympiad, 5
Let, $ a,b,c$ real positive numbers with $ abc \equal{} 1$
Prove:
$ (\frac {a}{1 \plus{} ab})^2 \plus{} (\frac {b}{1 \plus{} bc})^2 \plus{} (\frac {c}{1 \plus{} ca})^2\geq \frac {3}{4}$
Thanks!
2009 Math Prize For Girls Problems, 13
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?