Found problems: 85335
Geometry Mathley 2011-12, 14.3
Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$.
Nguyá»…n Minh HÃ
2016 Purple Comet Problems, 20
Positive integers a, b, c, d, and e satisfy the equations
$$(a + 1)(3bc + 1) = d + 3e + 1$$
$$(b + 1)(3ca + 1) = 3d + e + 13$$
$$(c + 1)(3ab + 1) = 4(26-d- e) - 1$$
Find $d^2+e^2$.
2014 BMT Spring, 7
If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$, then determine $|a + b|$.
2014 AMC 10, 24
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
$ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $
2005 Taiwan TST Round 2, 4
A quadrilateral $PQRS$ has an inscribed circle, the points of tangencies with sides $PQ$, $QR$, $RS$, $SP$ being $A$, $B$, $C$, $D$, respectively. Let the midpoints of $AB$, $BC$, $CD$, $DA$ be $E$, $F$, $G$, $H$, respectively. Prove that the angle between segments $PR$ and $QS$ is equal to the angle between segments $EG$ and $FH$.
1999 AIME Problems, 6
A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k.$
2011 Morocco National Olympiad, 4
Let $a, b, c, d, m, n$ be positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}=1989$, $n^{2}=max\left \{ a,b,c,d \right \}$ and $a+b+c+d=m^{2}$.
Find the values of $m$ and $n$.
2016 ASDAN Math Tournament, 8
Let $ABC$ be a triangle with $AB=24$, $BC=30$, and $AC=36$. Point $M$ lies on $BC$ such that $BM=12$, and point $N$ lies on $AC$ such that $CN=20$. Let $X$ be the intersection of $AM$ and $BN$ and let line $CX$ intersect $AB$ at point $L$. Compute
$$\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.$$
2010 Contests, 524
Evaluate the following definite integral.
\[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]
1959 AMC 12/AHSME, 48
Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is:
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $
1990 Putnam, B6
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
Kyiv City MO Juniors 2003+ geometry, 2012.9.5
The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$.
(Nagel Igor)
1998 Swedish Mathematical Competition, 3
A cube side $5$ is made up of unit cubes. Two small cubes are [i]adjacent [/i] if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).
2021 Belarusian National Olympiad, 8.1
Prove that there exists a $2021$-digit positive integer $\overline{a_1a_2\ldots a_{2021}}$, with all its digits being non-zero, such that for every $1 \leq n \leq 2020$ the following equality holds
$$\overline{a_1a_2\ldots a_n} \cdot \overline{a_{n+1}a_{n+2}\ldots a_{2021}}=\overline{a_1a_2\ldots a_{2021-n}} \cdot \overline{a_{2022-n}a_{2023-n}\ldots a_{2021}}$$
and all four numbers in the equality are pairwise different.
2007 Harvard-MIT Mathematics Tournament, 3
Let $a$ be a positive real number. Find the value of $a$ such that the definite integral \[\int_a^{a^2} \dfrac{dx}{x+\sqrt{x}}\] achieves its smallest possible value.
2008 AIME Problems, 10
The diagram below shows a $ 4\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors.
[asy]unitsize(0.25inch);
defaultpen(linewidth(0.7));
int i, j;
for(i = 0; i < 4; ++i)
for(j = 0; j < 4; ++j)
dot(((real)i, (real)j));[/asy]Define a [i]growing path[/i] to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $ m$ be the maximum possible number of points in a growing path, and let $ r$ be the number of growing paths consisting of exactly $ m$ points. Find $ mr$.
2010 Oral Moscow Geometry Olympiad, 1
Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$.
[img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]
Estonia Open Senior - geometry, 2004.1.3
a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions
(1) $ABCD$ is not cyclic;
(2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths;
(3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal?
b) Does there exist such a non-convex quadrangle?
2012 China National Olympiad, 2
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?
2005 IberoAmerican, 6
Let $n$ be a fixed positive integer. The points $A_1$, $A_2$, $\ldots$, $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint circumferences, each with diameter $A_iA_j$ for some $i \neq j$ and such that every point $A_k$ belongs to exactly one circumference. Points in the same circumference must be of the same color.
Determine the number of ways of coloring these $2n$ points when we vary the $n$ circumferences and the distribution of the colors.
2021 AMC 12/AHSME Spring, 7
Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?
$\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$
2020 Iran Team Selection Test, 1
We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.
[i]Proposed by Masud Shafaie[/i]
1953 Polish MO Finals, 5
From point $ O $ a car starts on a straight road and travels with constant speed $ v $. A cyclist who is located at a distance $ a $ from point $ O $ and at a distance $ b $ from the road wants to deliver a letter to this car. What is the minimum speed a cyclist should ride to reach his goal?
2013 National Olympiad First Round, 17
Let $ABC$ be an equilateral triangle with side length $10$ and $P$ be a point inside the triangle such that $|PA|^2+ |PB|^2 + |PC|^2 = 128$. What is the area of a triangle with side lengths $|PA|,|PB|,|PC|$?
$
\textbf{(A)}\ 6\sqrt 3
\qquad\textbf{(B)}\ 7 \sqrt 3
\qquad\textbf{(C)}\ 8 \sqrt 3
\qquad\textbf{(D)}\ 9 \sqrt 3
\qquad\textbf{(E)}\ 10 \sqrt 3
$
2016 AMC 10, 3
For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together?
$\textbf{(A)}\ \$37.50 \qquad\textbf{(B)}\ \$50.00\qquad\textbf{(C)}\ \$87.50\qquad\textbf{(D)}\ \$90.00\qquad\textbf{(E)}\ \$92.50$