Found problems: 85335
2005 Paraguay Mathematical Olympiad, 3
The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$ Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .
1969 IMO Shortlist, 31
$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$
2000 Kurschak Competition, 2
Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$, $B_T$ and $C_T$ as the points where lines $AT$, $BT$, and $CT$ meet the circumcircle of $ABC$. Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$.
2021 LMT Spring, A21 B22
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
In how many ways
Can you add three integers
Summing seventeen?
Order matters here.
For example, eight, three, six
Is not eight, six, three.
All nonnegative,
Do not need to be distinct.
What is your answer?
[i]Proposed by Derek Gao[/i]
2019 Jozsef Wildt International Math Competition, W. 46
Let $x$, $y$, $z > 0$ such that $x^2 + y^2 + z^2 = 3$. Then $$x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}$$
1997 Croatia National Olympiad, Problem 2
Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum
$$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.
1992 All Soviet Union Mathematical Olympiad, 574
Let $$f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$$, where $a, b, c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros.
1999 AMC 8, 14
In trapezoid $ABCD$ , the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is
[asy]
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);
draw((4,3)--(4,0),dashed);
draw((3.2,0)--(3.2,.8)--(4,.8));
label("$A$",(0,0),SW);
label("$B$",(4,3),NW);
label("$C$",(12,3),NE);
label("$D$",(16,0),SE);
label("$8$",(8,3),N);
label("$16$",(8,0),S);
label("$3$",(4,1.5),E);[/asy]
$ \text{(A)}\ 27\qquad\text{(B)}\ 30\qquad\text{(C)}\ 32\qquad\text{(D)}\ 34\qquad\text{(E)}\ 48 $
1966 IMO Longlists, 28
In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of
[b]a.)[/b] all vertices $A$ of such triangles;
[b]b.)[/b] all vertices $B$ of such triangles;
[b]c.)[/b] all vertices $C$ of such triangles.
2004 IMC, 2
Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have
\[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \]
Prove that
\[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]
2024 IFYM, Sozopol, 6
Each of 9 girls participates in several (one or more) theater groups, so that there are no two identical groups. Each of them is randomly assigned a positive integer between 1 and 30 inclusive. We call a group \textit{small} if the sum of the numbers of its members does not exceed the sum of any other group. Prove that regardless of which girl participates in which group, the probability that after receiving the numbers there will be a unique small group is at least \( \frac{7}{10} \).
2010 Contests, 3
We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]
2008 Harvard-MIT Mathematics Tournament, 1
Determine all pairs $ (a,b)$ of real numbers such that $ 10, a, b, ab$ is an arithmetic progression.
2004 Romania Team Selection Test, 11
Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.
[i]Alternative formulation.[/i] The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.
2012 National Olympiad First Round, 34
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$, what is the least integer $n\geq 2012$?
$ \textbf{(A)}\ 2012 \qquad \textbf{(B)}\ 2013 \qquad \textbf{(C)}\ 2014 \qquad \textbf{(D)}\ 2015 \qquad \textbf{(E)}\ 2016$
2005 Gheorghe Vranceanu, 4
Let be a triangle $ ABC $ and the points $ E,F,M,N $ positioned in this way: $ E,F $ on the segment $ BC $ (excluding its endpoints), $ M $ on the segment $ AC $ (excluding its endpoints) and $ N $ on the segment $ AC $ (excluding its endpoints). Knowing that $ BAE $ is similar to $ FAC $ and that $ BE=BM,FC=CN,AM=AN, $ show that $ ABC $ is isosceles.
2021 ABMC., 2021 Nov
[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched to Allstate?
[b]p2.[/b] The $7$-digit number $N$ can be written as $\underline{A} \,\, \underline{2} \,\,\underline{0} \,\,\underline{B} \,\,\underline{2} \,\, \underline{1} \,\,\underline{5}$. How many values of $N$ are divisible by $9$?
[b]p3.[/b] The solutions to the equation $x^2-18x-115 = 0$ can be represented as $a$ and $b$. What is $a^2+2ab+b^2$?
[b]p4.[/b] The exterior angles of a regular polygon measure to $4$ degrees. What is a third of the number of sides of this polygon?
[b]p5.[/b] Charlie Brown is having a thanksgiving party.
$\bullet$ He wants one turkey, with three different sizes to choose from.
$\bullet$ He wants to have two or three vegetable dishes, when he can pick from Mashed Potatoes, Saut´eed Brussels Sprouts, Roasted Butternut Squash, Buttery Green Beans, and Sweet Yams;
$\bullet$ He wants two desserts out of Pumpkin Pie, Apple Pie, Carrot Cake, and Cheesecake.
How many different combinations of menus are there?
[b]p6.[/b] In the diagram below, $\overline{AD} \cong \overline{CD}$ and $\vartriangle DAB$ is a right triangle with $\angle DAB = 90^o$. Given that the radius of the circle is $6$ and $m \angle ADC = 30^o$, if the length of minor arc $AB$ is written as $a\pi$, what is $a$?
[img]https://cdn.artofproblemsolving.com/attachments/d/9/ea57032a30c16f4402886af086064261d6828b.png[/img]
[b]p7.[/b] This Halloween, Owen and his two friends dressed up as guards from Squid Game. They needed to make three masks, which were black circles with a white equilateral triangle, circle, or square inscribed in their upper halves. Resourcefully, they used black paper circles with a radius of $5$ inches and white tape to create these masks. Ignoring the width of the tape, how much tape did they use? If the length can be expressed $a\sqrt{b}+c\sqrt{d}+ \frac{e}{f} \pi$ such that $b$ and $d$ are not divisible by the square of any prime, and $e$ and $f$ are relatively prime, find $a + b + c + d + e + f$.
[img]https://cdn.artofproblemsolving.com/attachments/0/c/bafe3f9939bd5767ba5cf77a51031dd32bbbec.png[/img]
[b]p8.[/b] Given $LCM (10^8, 8^{10}, n) = 20^{15}$, where $n$ is a positive integer, find the total number of possible values of $n$.
[b]p9.[/b] If one can represent the infinite progression $\frac{1}{11} + \frac{2}{13} + \frac{3}{121} + \frac{4}{169} + \frac{5}{1331} + \frac{6}{2197}+ ...$ as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, what is $a$?
[b]p10.[/b] Consider a tiled $3\times 3$ square without a center tile. How many ways are there to color the squares such that no two colored squares are adjacent (vertically or horizontally)? Consider rotations of an configuration to be the same, and consider the no-color configuration to be a coloring.
[b]p11.[/b] Let $ABC$ be a triangle with $AB = 4$ and $AC = 7$. Let $AD$ be an angle bisector of triangle $ABC$. Point $M$ is on $AC$ such that $AD$ intersects $BM$ at point $P$, and $AP : PD = 3 : 1$. If the ratio $AM : MC$ can be expressed as $\frac{a}{b}$ such that $a$, $b$ are relatively prime positive integers, find $a + b$.
[b]p12.[/b] For a positive integer $n$, define $f(n)$ as the number of positive integers less than or equal to $n$ that are coprime with $n$. For example, $f(9) = 6$ because $9$ does not have any common divisors with $1$, $2$, $4$, $5$, $7$, or $8$. Calculate: $$\sum^{100}_{i=2} \left( 29^{f(i)}\,\,\, mod \,\,i \right).$$
[b]p13.[/b] Let $ABC$ be an equilateral triangle. Let $P$ be a randomly selected point in the incircle of $ABC$. Find $a+b+c+d$ if the probability that $\angle BPC$ is acute can be expressed as $\frac{a\sqrt{b} -c\pi}{d\pi }$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c, d) = 1$ and $b$ is not divisible by the square of any prime.
[b]p14.[/b] When the following expression is simplified by expanding then combining like terms, how many terms are in the resulting expression? $$(a + b + c + d)^{100} + (a + b - c - d)^{100}$$
[b]p15.[/b] Jerry has a rectangular box with integral side lengths. If $3$ units are added to each side of the box, the volume of the box is tripled. What is the largest possible volume of this box?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2007 Harvard-MIT Mathematics Tournament, 10
Let $A_{12}$ denote the answer to problem $12$. There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}>B>C>D>0$ and \[\overline{A_{12}BCD}-\overline{DCBA_{12}}=\overline{BDA_{12}C},\] where $\overline{A_{12}BCD}$ denotes the four digit base $10$ integer. Compute $B+C+D$.
2021 Kurschak Competition, 1
Let $P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)$ be points on the plane such that $P_0P_1P_2\Delta$ contains the origin $O$. Show that the areas of triangles $P_0OP_1,P_0OP_2,P_1OP_2$ form a geometric sequence in that order if and only if there exists a real number $x$, such that
$$
a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0
$$
MBMT Guts Rounds, 2015.1
Mr. Stein is ordering a two-course dessert at a restaurant. For each course, he can choose to eat pie, cake, rødgrød, and crème brûlée, but he doesn't want to have the same dessert twice. In how many ways can Mr. Stein order his meal? (Order matters.)
1971 Czech and Slovak Olympiad III A, 1
Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that
\begin{align*}
cy-bz &\ge 0, \\
az-cx &\ge 0, \\
bx-ay &\ge 0.
\end{align*}
2015 JBMO Shortlist, 4
Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]
2012 Tuymaada Olympiad, 2
Solve in positive integers the following equation:
\[{1\over n^2}-{3\over 2n^3}={1\over m^2}\]
[i]Proposed by A. Golovanov[/i]
2009 Today's Calculation Of Integral, 506
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.
2021 Purple Comet Problems, 22
The least positive angle $\alpha$ for which $$\left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256}$$ has a degree measure of $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.