Found problems: 85335
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.
2015 IMC, 5
Let $n\ge2$, let $A_1,A_2,\ldots,A_{n+1}$ be $n+1$ points in the
$n$-dimensional Euclidean space, not lying on the same hyperplane,
and let $B$ be a point strictly inside the convex hull of
$A_1,A_2,\ldots,A_{n+1}$. Prove that $\angle A_iBA_j>90^\circ$ holds
for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i<j\le
n+1}$.
Proposed by Géza Kós, Eötvös University, Budapest
2018 PUMaC Number Theory A, 6
Find the remainder of
$$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$
when divided by $101$.
2021 BMT, T3
Let $N$ be the number of tuples $(a_1, a_2,..., a_{150})$ satisfying:
$\bullet$ $a_i \in \{2, 3, 5, 7, 11\}$ for all $1 \le i \le 99$.
$\bullet$ $a_i \in \{2, 4, 6, 8\}$ for all $100 \le i \le 150$.
$\bullet$ $\sum^{150}_{i=1}a_i$ is divisible by $8$.
Compute the last three digits of $N$.
2024 Caucasus Mathematical Olympiad, 1
Let $a, b, c, d$ be positive real numbers. It is given that at least one of the following two conditions holds:
$$ab >\min(\frac{c}{d}, \frac{d}{c}), cd >\min(\frac{a}{b}, \frac{b}{a}).$$ Show that at least one of the following two conditions holds: $$bd>\min(\frac{c}{a}, \frac{a}{c}), ca >\min(\frac{d}{b}, \frac{b}{d}).$$
2011 Bundeswettbewerb Mathematik, 3
The diagonals of a convex pentagon divide each of its interior angles into three equal parts.
Does it follow that the pentagon is regular?
2000 AMC 8, 10
Ara and Shea were once the same height. Since then Shea has grown $20\%$ while Ara has grow half as many inches as Shea. Shea is now $60$ inches tall. How tall, in inches, is Ara now?
$\text{(A)}\ 48 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$
2021 AIME Problems, 13
Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.
2008 Iran Team Selection Test, 3
Suppose that $ T$ is a tree with $ k$ edges. Prove that the $ k$-dimensional cube can be partitioned to graphs isomorphic to $ T$.
2014 Harvard-MIT Mathematics Tournament, 10
[6] Find the number of sets $\mathcal{F}$ of subsets of the set $\{1,\ldots,2014\}$ such that:
a) For any subsets $S_1,S_2 \in \mathcal{F}, S_1 \cap S_2 \in \mathcal{F}$.
b) If $S \in \mathcal{F}$, $T \subseteq \{1,\ldots,2014\}$, and $S \subseteq T$, then $T \in \mathcal{F}$.
2019 Baltic Way, 14
Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.
2018 ELMO Shortlist, 1
Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$.
[i]Proposed by Ankan Bhattacharya[/i]
2003 Iran MO (3rd Round), 15
Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$.
( for example the diffrence of this two row is only in one index
110
100)
[i]Edited by Myth[/i]
2001 Moldova National Olympiad, Problem 6
Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.
Kvant 2023, M2754
Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root.
Proposed by N. Agakhanov
2008 Princeton University Math Competition, A9/B10
How many spanning trees does the following graph (with $6$ vertices and $9$ edges) have?
(A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.)
[img]https://cdn.artofproblemsolving.com/attachments/0/4/0e53e0fbb141b66a7b1c08696be2c5dfe68067.png[/img]
2022 Czech-Austrian-Polish-Slovak Match, 3
Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$. A variable line $l$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$, and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$. Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$. The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$. Prove that line $RT$/is tangent to a fixed circle, independent of the choice of $l$.
2002 AMC 10, 18
For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$
2002 Singapore Senior Math Olympiad, 2
The vertices of a triangle inscribed in a circle are the points of tangency of a triangle circumscribed about the circle. Prove that the product of the perpendicular distances from any point on the circle to the sides of the inscribed triangle is the same as the product of the perpendicular distances from the same point to the sides of the circumscribed triangle.