Found problems: 85335
2022 CHMMC Winter (2022-23), 4
Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, $CA = 6$. Triangles $APB$ and $CQA$ are erected outside $ABC$ such that $AP=PB$, $\overline{AP}\perp \overline{PB}$ and $CQ=QA$, $\overline{CQ}\perp \overline{QA}$. Pick a point $X$ uniformly at random from segment $\overline{BC}$. What is the expected value of the area of triangle $PXQ$?
2010 AMC 10, 22
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
$ \textbf{(A)}\ 28 \qquad
\textbf{(B)}\ 56 \qquad
\textbf{(C)}\ 70 \qquad
\textbf{(D)}\ 84 \qquad
\textbf{(E)}\ 140$
2016 India IMO Training Camp, 3
Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.
2017 ASDAN Math Tournament, 3
Compute
$$\int_0^1\frac{x^{2017}-1}{\log x}dx.$$
2017 South East Mathematical Olympiad, 8
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$
Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and
$$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$
Determine the largest possible number of elements in $A$.
2010 Today's Calculation Of Integral, 605
Let $f(x)$ be a differentiable function. Find the following limit value:
\[\lim_{n\to\infty} \dbinom{n}{k}\left\{f\left(\frac{x}{n}\right)-f(0)\right\}^k.\]
Especially, for $f(x)=(x-\alpha)(x-\beta)$ find the limit value above.
1956 Tokyo Institute of Technology entrance exam
2025 Sharygin Geometry Olympiad, 13
Each two opposite sides of a convex $2n$-gon are parallel. (Two sides are opposite if one passes $n-1$ other sides moving from one side to another along the borderline of the $2n$-gon.) The pair of opposite sides is called regular if there exists a common perpendicular to them such that its endpoints lie on the sides and not on their extensions. Which is the minimal possible number of regular pairs?
Proposed by: B.Frenkin
1994 Turkey Team Selection Test, 1
$f$ is a function defined on integers and satisfies $f(x)+f(x+3)=x^2$ for every integer $x$. If $f(19)=94$, then calculate $f(94)$.
2018 Sharygin Geometry Olympiad, 2
A triangle $ABC$ is given. A circle $\gamma$ centered at $A$ meets segments $AB$ and $AC$. The common chord of $\gamma$ and the circumcircle of $ABC$ meets $AB$ and $AC$ at $X$ and $Y$, respectively. The segments $CX$ and $BY$ meet $\gamma$ at point $S$ and $T$, respectively. The circumcircles of triangles $ACT$ and $BAS$ meet at points $A$ and $P$. Prove that $CX, BY$ and $AP$ concur.
2011 Turkey MO (2nd round), 5
Let $M$ and $N$ be two regular polygonic area.Define $K(M,N)$ as the midpoints of segments $[AB]$ such that $A$ belong to $M$ and $B$ belong to $N$. Find all situations of $M$ and $N$ such that $K(M,N)$ is a regualr polygonic area too.
2016 Tuymaada Olympiad, 5
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
1995 IMC, 8
Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$,
$b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$. Calculate
$$\sum_{n=1}^{\infty}b_{n}2^{n}.$$
2004 Harvard-MIT Mathematics Tournament, 3
A swimming pool is in the shape of a circle with diameter $60$ ft. The depth varies linearly along the east-west direction from $3$ ft at the shallow end in the east to $15$ ft at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at all along the north-south direction. What is the volume of the pool, in ft$^3$?
2011 Romanian Master of Mathematics, 1
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.
[i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]
2022 China Girls Math Olympiad, 2
Let $n$ be a positive integer. There are $3n$ women's volleyball teams in the tournament, with no more than one match between every two teams (there are no ties in volleyball). We know that there are $3n^2$ games played in this tournament.
Proof: There exists a team with at least $\frac{n}{4}$ win and $\frac{n}{4}$ loss
LMT Speed Rounds, 11
Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees.
[i]Proposed by Edwin Zhao[/i]
2005 IMAR Test, 1
The incircle of triangle $ABC$ touches the sides $BC,CA,AB$ at the points $D,E,F$, respectively. Let $K$ be a point on the side $BC$ and let $M$ be the point on the line segment $AK$ such that $AM=AE=AF$. Denote by $L,N$ the incenters of triangles $ABK,ACK$, respectively.
Prove that $K$ is the foot of the altitude from $A$ if and only if $DLMN$ is a square.
[hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2[/url]
[/hide]
[i]Bogdan Enescu[/i]
MathLinks Contest 4th, 4.2
We say that two triangles $T_1$ and $T_2$ are contained one in each other, and we write $T_1 \subset T_2$, if and only if all the points of the triangle $T_1$ lie on the sides or in the interior of the triangle $T_2$.
Let $\Delta$ be a triangle of area $S$, and let $\Delta_1 \subset \Delta$ be the largest equilateral triangle with this property, and let $\Delta \subset \Delta_2$ be the smallest equilateral triangle with this property (in terms of areas). Let $S_1, S_2$ be the areas of $\Delta_1, \Delta_2$ respectively.
Prove that $S_1S_2 = S^2$.
Bonus question: : Does this statement hold for quadrilaterals (and squares)?
2010 CHMMC Fall, 14
A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by $(a, b, c, d)$, where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given
by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional
volume of the solid formed by the intersection.
2018 AIME Problems, 4
In equiangular octagon $CAROLINE$, $CA = RO = LI = NE = \sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, that total area of the six triangular regions. Then $K=\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
2007 iTest Tournament of Champions, 3
For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$?
2021 Indonesia TST, N
Let $n$ be a positive integer. Prove that $$\gcd(\underbrace{11\dots 1}_{n \text{times}},n)\mid 1+10^k+10^{2k}+\dots+10^{(n-1)k}$$ for all positive integer $k$.
2010 USAJMO, 5
Two permutations $a_1,a_2,\dots,a_{2010}$ and $b_1,b_2,\dots,b_{2010}$ of the numbers $1,2,\dots,2010$ are said to [i]intersect[/i] if $a_k=b_k$ for some value of $k$ in the range $1\le k\le 2010$. Show that there exist $1006$ permutations of the numbers $1,2,\dots,2010$ such that any other such permutation is guaranteed to intersect at least one of these $1006$ permutations.
1966 IMO Longlists, 38
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
1993 Italy TST, 2
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$.