Found problems: 85335
1991 Putnam, A1
The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).
2011 AMC 12/AHSME, 5
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 60
$
2000 Croatia National Olympiad, Problem 2
The incircle of a triangle $ABC$ touches $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Find the angles of $\triangle A_1B_1C_1$ in terms of the angles of $\triangle ABC$.
1969 Putnam, B4
Show that any curve of unit length can be covered by a closed rectangle of area $1 \slash 4$.
2019 Jozsef Wildt International Math Competition, W. 24
If $a$, $b$, $c > 0$, prove that$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}$$
MBMT Guts Rounds, 2015.16
Your math teacher asks you to rationalize the denominator of the expression $\frac{a}{b + \sqrt{c}}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. You find that $\frac{a}{b + \sqrt{c}}$ is equal to $\frac{30 - 5\sqrt{14}}{11}$. Compute the triple $(a,b,c)$.
2022 Latvia Baltic Way TST, P13
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square.
Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.
2015 Abels Math Contest (Norwegian MO) Final, 2b
Nils is playing a game with a bag originally containing $n$ red and one black marble.
He begins with a fortune equal to $1$. In each move he picks a real number $x$ with $0 \le x \le y$, where his present fortune is $y$. Then he draws a marble from the bag. If the marble is red, his fortune increases by $x$, but if it is black, it decreases by $x$. The game is over after $n$ moves when there is only a single marble left.
In each move Nils chooses $x$ so that he ensures a final fortune greater or equal to $Y$ .
What is the largest possible value of $Y$?
2004 Iran MO (3rd Round), 9
Let $ABC$ be a triangle, and $O$ the center of its circumcircle.
Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively.
Prove that $\measuredangle ROS=\measuredangle BAC$.
2006 Italy TST, 1
The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively.
a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel.
b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.
2020 MBMT, 21
Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with \$256, find the probability that after 8 bets, he will have at least \$50.
[i]Proposed by Jeffrey Tong[/i]
2005 Putnam, A6
Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?
2014 India PRMO, 3
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?
1961 All Russian Mathematical Olympiad, 010
Nicholas and Peter are dividing $(2n+1)$ nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common.
1 stage: Peter divides nuts onto $2$ heaps, each contain not less than $2$ nuts.
2 stage: Nicholas divides both heaps onto $2$ heaps, each contain not less than $1$ nut.
3 stage:
1 way: Nicholas takes the biggest and the least heaps.
2 way: Nicholas takes two middle size heaps.
3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice.
Find the most and the least profitable method for the Nicholas.
2025 Kosovo National Mathematical Olympiad`, P4
Show that for any real numbers $a$ and $b$ different from $0$, the inequality
$$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$
holds. When is equality achieved?
2012 ELMO Shortlist, 6
Prove that if $a$ and $b$ are positive integers and $ab>1$, then
\[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.
[i]Calvin Deng.[/i]
2014 BMT Spring, P2
Given an integer $n\ge2$, the graph $G$ is defined by:
- Vertices of $G$ are represented by binary strings of length $n$
- Two vertices $a,b$ are connected by an edge if and only if they differ in exactly $2$ places
Let $S$ be a subset of the vertices of $G$, and let $S'$ be the set of edges between vertices in $S$ and vertices not in $S$. Show that if $|S|$ (the size of $S$) $\le2^{n-2}$, then $|S'|\ge|S|$.
2006 National Olympiad First Round, 35
$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$?
$
\textbf{(A)}\ -3
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ 2\sqrt 3
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \text{None of above}
$
2014 Estonia Team Selection Test, 2
Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$
2017 Junior Regional Olympiad - FBH, 5
Find all positive integers $a$ and $b$ such that number $p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is rational number
1996 AMC 8, 24
The measure of angle $ABC$ is $50^\circ $, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is
[asy]
pair A,B,C,D;
A = (0,0); B = (9,10); C = (10,0); D = (6.66,3);
dot(A); dot(B); dot(C); dot(D);
draw(A--B--C--cycle);
draw(A--D--C);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,N);
label("$50^\circ $",(9.4,8.8),SW);
[/asy]
$\text{(A)}\ 90^\circ \qquad \text{(B)}\ 100^\circ \qquad \text{(C)}\ 115^\circ \qquad \text{(D)}\ 122.5^\circ \qquad \text{(E)}\ 125^\circ $
2016 Math Prize for Girls Problems, 20
Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be random integers chosen independently and uniformly from the set $\{ 0, 1, 2, \dots, 23 \}$. (Note that the integers are not necessarily distinct.) Find the probability that
\[
\sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0.
\]
(Here $\operatorname{cis} \theta$ means $\cos \theta + i \sin \theta$.)
2014 VTRMC, Problem 5
Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.
2020 Sharygin Geometry Olympiad, 10
Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$. Call the link [i]good[/i], if it touches $\omega$, and [i]bad[/i] otherwise (i.e. if the extension of this link touches $\omega$). Prove that the number of bad links is even.
2017 Iran Team Selection Test, 5
Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as
$$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$
Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that
$$P_{2n}(x)=P_n(x^2+c).$$
[i]Proposed by Navid Safaei[/i]