Found problems: 85335
2021 Philippine MO, 5
A positive integer is called $\emph{lucky}$ if it is divisible by $7$, and the sum of its digits is also divisible by $7$. Fix a positive integer $n$. Show that there exists some lucky integer $l$ such that $\left|n - l\right| \leq 70$.
2025 NCMO, 2
In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent.
[i]Alan Cheng[/i]
2021 Latvia TST, 2.2
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
2019 Durer Math Competition Finals, 8
A chess piece is placed on one of the squares of an $8\times 8$ chessboard where it begins a tour of the board: it moves from square to square, only moving horizontally or vertically. It visits every square precisely once, and ends up exactly where it started. What is the maximum number of times the piece can change direction along its tour?
2007 Moldova Team Selection Test, 4
Consider a convex polygon $A_{1}A_{2}\ldots A_{n}$ and a point $M$ inside it. The lines $A_{i}M$ intersect the perimeter of the polygon second time in the points $B_{i}$. The polygon is called balanced if all sides of the polygon contain exactly one of points $B_{i}$ (strictly inside). Find all balanced polygons.
[Note: The problem originally asked for which $n$ all convex polygons of $n$ sides are balanced. A misunderstanding made this version of the problem appear at the contest]
2017 Saudi Arabia IMO TST, 1
Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \in AC, F \in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\angle OKT = 90^o$.
2005 Italy TST, 2
$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality.
$(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.
2021 AMC 10 Fall, 9
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{3}{13}\qquad\textbf{(C) }\frac{7}{27}\qquad\textbf{(D) }\frac{2}{7}\qquad\textbf{(E) }\frac{1}{3}$
1998 All-Russian Olympiad Regional Round, 8.3
There are 52 cards in the deck, 13 of each suit. Vanya draws from the deck one card at a time. Removed cards are not returned to the deck. Every time Before taking out the card, Vanya makes a wish for some suit.Prove that if Vanya makes a wish every time, , the cards of which are in the deck has no less cards left than cards of any other suit, then the hidden suit will fall with the suit of the card drawn at least 13 times.
1966 IMO Longlists, 12
Find digits $x, y, z$ such that the equality
\[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\]
holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.
2009 Germany Team Selection Test, 3
There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that
\[
\angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ}
\]
if and only if the diagonals $ AC$ and $ BD$ are perpendicular.
[i]Proposed by Dusan Djukic, Serbia[/i]
1985 Federal Competition For Advanced Students, P2, 4
Find all natural numbers $ n$ such that the equation:
$ a_{n\plus{}1} x^2\minus{}2x \sqrt{a_1^2\plus{}a_2^2\plus{}...\plus{}a_{n\plus{}1}^2}\plus{}a_1\plus{}a_2\plus{}...\plus{}a_n\equal{}0$
has real solutions for all real numbers $ a_1,a_2,...,a_{n\plus{}1}$.
2016 Thailand TSTST, 2
Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.
1997 Pre-Preparation Course Examination, 1
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
2021 AMC 12/AHSME Fall, 17
A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?
$\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\
\frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}$
2015 USAMO, 6
Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)
1976 Kurschak Competition, 1
$ABCD$ is a parallelogram. $P$ is a point outside the parallelogram such that angles $\angle PAB$ and $\angle PCB$ have the same value but opposite orientation. Show that $\angle APB = \angle DPC$.
OMMC POTM, 2022 6
Let $G$ be the centroid of $\triangle ABC.$ A rotation $120^\circ$ clockwise about $G$ takes $B$ and $C$ to $B_1$ and $C_1$ respectively. A rotation $120^\circ$ counterclockwise about $G$ takes $B$ and $C$ to $B_2$ and $C_2$ respectively. Prove $\triangle AB_1C_2$ and $\triangle AB_2C_1$ are equilateral.
[i]Proposed by Evan Chang (squareman), USA [/i]
[img]https://cdn.artofproblemsolving.com/attachments/3/b/46b4f09edcf17755df2dea3546881475db6eff.png[/img]
Today's calculation of integrals, 867
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
1993 Putnam, A5
Let U be the set formed as the union of three open intervals, $U = (-100, -10) \cup (1/101, 1/11) \cup (101/100, 11/10)$. Show that $\int_{U} \frac{(x^2-x)^2}{(x^3-3x+1)^2} dx$ is rational.
2025 Spain Mathematical Olympiad, 4
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$, satisfying $AB<AC$. The tangent line at $A$ to the circumcicle of $ABC$ intersects $BC$ in $T$. Let $X$ be the midpoint of $AH$. Prove that $\angle ATX=\angle OTB$.
2019 IMO Shortlist, N5
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
2016 Switzerland - Final Round, 1
Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.
2014 Korea - Final Round, 2
Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.
2022 Thailand TSTST, 1
Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.