Found problems: 275
1983 USAMO, 5
Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.
1972 USAMO, 1
The symbols $ (a,b,\ldots,g)$ and $ [a,b,\ldots,g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $ a,b,\ldots,g$. For example, $ (3,6,18)\equal{}3$ and $ [6,15]\equal{}30$. Prove that \[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}\equal{}\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]
1984 USAMO, 3
$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.
2021 Indonesia TST, G
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
1985 USAMO, 2
Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.
2012 USAJMO, 6
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
2017 USAMO, 4
Let $P_1$, $P_2$, $\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.
1978 USAMO, 1
Given that $a,b,c,d,e$ are real numbers such that
$a+b+c+d+e=8$,
$a^2+b^2+c^2+d^2+e^2=16$.
Determine the maximum value of $e$.
1980 USAMO, 5
Prove that for numbers $a,b,c$ in the interval $[0,1]$, \[\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}+(1-a)(1-b)(1-c) \le 1.\]
2005 MOP Homework, 6
Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.
2010 USAMO, 5
Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let\[
S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)}
\]Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$, then $m - n$ is divisible by $p$.
1981 USAMO, 2
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
2013 USAMO, 4
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
2021 Indonesia TST, G
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
2006 AIME Problems, 11
A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.
2008 National Olympiad First Round, 2
For which value of $A$, does the equation $3m^2n = n^3 + A$ have a solution in natural numbers?
$
\textbf{(A)}\ 301
\qquad\textbf{(B)}\ 403
\qquad\textbf{(C)}\ 415
\qquad\textbf{(D)}\ 427
\qquad\textbf{(E)}\ 481
$
2011 USAJMO, 6
Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.
2023 USAJMO, 2
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
2010 USAMO, 4
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.
1980 USAMO, 1
A two-pan balance is innacurate since its balance arms are of different lengths and its pans are of different weights. Three objects of different weights $A$, $B$, and $C$ are each weighed separately. When placed on the left-hand pan, they are balanced by weights $A_1$, $B_1$, and $C_1$, respectively. When $A$ and $B$ are placed on the right-hand pan, they are balanced by $A_2$ and $B_2$, respectively. Determine the true weight of $C$ in terms of $A_1, B_1, C_1, A_2$, and $B_2$.
2005 Romania National Olympiad, 3
Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.
2010 USAMO, 2
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
2023 USAMO, 1
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
2010 USAJMO, 3
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
1978 USAMO, 5
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.