Found problems: 85335
2020 LMT Fall, A16
Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively, and intersect at points $M$ and $N$. The radii of $\omega_1$ and $\omega_2$ are $12$ and $15$, respectively, and $O_1O_2 = 18$. A point $X$ is chosen on segment $MN$. Line $O_1X$ intersects $\omega_2$ at points $A$ and $C$, where $A$ is inside $\omega_1$. Similarly, line $O_2X$ intersects $\omega_1$ at points $B$ and $D$, where $B$ is inside $\omega_2$. The perpendicular bisectors of segments $AB$ and $CD$ intersect at point $P$. Given that $PO_1 = 30$, find $PO_2^2$.
[i]Proposed by Andrew Zhao[/i]
1988 National High School Mathematics League, 10
Lengths of two sides of a rectangle are $\sqrt2,1$. The rectangle rotates a round around one of its diagonal. Find the volume of the revolved body.
2024 LMT Fall, 20
Henry places some rooks and some kings in distinct cells of a $2\times 8$ grid such that no two rooks attack each other and no two kings attack each other. Find the maximum possible number of pieces on the board.
(Two rooks [i]attack[/i] each other if they are in the same row or column and no pieces are between them. Two kings attack each other if their cells share a vertex.)
2024 Mexican University Math Olympiad, 6
Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that
\[
(p(x)^2)'' \leq K(p'(x))^2.
\]
2015 Korea Junior Math Olympiad, 4
Reals $a,b,c,x,y$ satisfy $a^2+b^2+c^2=x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$
2024 CCA Math Bonanza, L1.3
Find the number of $10$ digit palindromes that are not divisible by $11$.
[i]Lightning 1.3[/i]
2020 Iran RMM TST, 1
For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd.
[i]Proposed by Navid Safaie[/i]
1969 IMO Shortlist, 18
$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$
2021 Girls in Math at Yale, 11
A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism?
[i]Proposed by Deyuan Li and Andrew Milas[/i]
1997 AIME Problems, 6
Point $B$ is in the exterior of the regular $n$-sided polygon $A_1A_2\cdots A_n,$ and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_n, A_1,$ and $B$ are consecutive vertices of a regular polygon?
2011 Rioplatense Mathematical Olympiad, Level 3, 1
Given a positive integer $n$, an operation consists of replacing $n$ with either $2n-1$, $3n-2$ or $5n-4$. A number $b$ is said to be a [i]follower[/i] of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$.
2014 AMC 8, 14
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?
[asy]
size(250);
defaultpen(linewidth(0.8));
pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0);
draw(A--B--E--D--cycle^^C--D);
draw(rightanglemark(D,C,E,30));
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,S);
label("$D$",D,N);
label("$E$",E,S);
label("$5$",A/2,W);
label("$6$",(A+D)/2,N);
[/asy]
$\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$
2019 International Zhautykov OIympiad, 3
Triangle $ABC$ is given. The median $CM$ intersects the circumference of $ABC$ in $N$. $P$ and $Q$ are chosen on the rays $CA$ and $CB$ respectively, such that $PM$ is parallel to $BN$ and $QM$ is parallel to $AN$. Points $X$ and $Y$ are chosen on the segments $PM$ and $QM$ respectively, such that both $PY$ and $QX$ touch the circumference of $ABC$. Let $Z$ be intersection of $PY$ and $QX$. Prove that, the quadrilateral $MXZY$ is circumscribed.
2003 China Team Selection Test, 1
Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.
Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.41
Let $AC$ be the largest side of the triangle $ABC$. The point M is selected on the ray $AC$ ray, and point $N$ on ray $CA$ such that $CN = CB$ and$ AM = AB$ .
a) Prove that $\vartriangle ABC$ is isosceles if we know that $BM = BN$.
b) Will the statement remain true if $AC$ is not necessarily the largest side of triangle $ABC$?
2015 Saudi Arabia BMO TST, 3
Let $ABC$ be a triangle, $H_a, H_b$ and $H_c$ the feet of its altitudes from $A, B$ and $C$, respectively, $T_a, T_b, T_c$ its touchpoints of the incircle with the sides $BC, CA$ and $AB$, respectively. The circumcircles of triangles $AH_bH_c$ and $AT_bT_c$ intersect again at $A'$. The circumcircles of triangles $BH_cH_a$ and $BT_cT_a$ intersect again at $B'$. The circumcircles of triangles $CH_aH_b$ and $CT_aT_b$ intersect again at $C'$. Prove that the points $A',B',C'$ are collinear.
Malik Talbi
2023 LMT Fall, 8
Let $J$ , $E$, $R$, and $Y$ be four positive integers chosen independently and uniformly at random from the set of factors of $1428$. What is the probability that $JERRY = 1428$? Express your answer in the form $\frac{a}{b\cdot 2^n}$ where $n$ is a nonnegative integer, $a $and $b$ are odd, and gcd $(a,b) = 1$.
2011 AMC 10, 14
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$ \textbf{(A)}\ 52 \qquad
\textbf{(B)}\ 58 \qquad
\textbf{(C)}\ 62 \qquad
\textbf{(D)}\ 68 \qquad
\textbf{(E)}\ 70 $
1990 IMO Shortlist, 23
Determine all integers $ n > 1$ such that
\[ \frac {2^n \plus{} 1}{n^2}
\]
is an integer.
1988 All Soviet Union Mathematical Olympiad, 469
If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.
1988 IMO Longlists, 24
Find the positive integers $x_1, x_2, \ldots, x_{29}$ at least one of which is greater that 1988 so that
\[ x^2_1 + x^2_2 + \ldots x^2_{29} = 29 \cdot x_1 \cdot x_2 \ldots x_{29}. \]
2002 Baltic Way, 5
Find all pairs $(a,b)$ of positive rational numbers such that
\[\sqrt{a}+\sqrt{b}=\sqrt{2+\sqrt{3}}. \]
2007 Indonesia TST, 4
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.
2011 ELMO Shortlist, 4
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]
2005 MOP Homework, 3
Points $M$ and $M'$ are isogonal conjugates in the traingle $ABC$. We draw perpendiculars $MP$, $MQ$, $MR$, and $M'P'$, $M'Q'$, $M'R'$ to the sides $BC$, $AC$, $AB$ respectively. Let $QR$, $Q'R'$, and $RP$, $R'P'$ and $PQ$, $P'Q'$ intersect at $E$, $F$, $G$ respectively. Show that the lines $EA$, $FB$, and $GC$ are parallel.