Found problems: 275
2005 Romania National Olympiad, 1
Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that:
a) $DE=AF$;
b) $AD\cdot AB = DE\cdot DM$.
[i]Daniela and Marius Lobaza, Timisoara[/i]
2011 Indonesia TST, 2
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
1981 USAMO, 4
The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.
$\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.
2018 USAMO, 2
Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that \[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\] for all $x,y,z >0$ with $xyz =1$.
1998 USAMO, 6
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)
2018 USAMO, 4
Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers
\[a_1 + k, a_2 + 2k, \dots, a_p + pk\]
produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$.
[i]Proposed by Ankan Bhattacharya[/i]
2010 USAMO, 3
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
1982 USAMO, 4
Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.
2012 Math Prize For Girls Problems, 8
Suppose that $x$, $y$, and $z$ are real numbers such that $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$. What is the largest possible value of $z$?
2025 USAMO, 3
Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds:
[center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center]
[center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center]
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.
[i]Note:[/i] $\triangle UVW$ is [i]directly similar[/i] to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.
2010 AIME Problems, 10
Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.
1980 USAMO, 4
The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.
2023 USAMO, 4
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
2015 USA Team Selection Test, 3
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?
[i]Proposed by Linus Hamilton[/i]
2004 India IMO Training Camp, 2
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
1986 USAMO, 3
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
2025 USAMO, 1
Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.
2025 USAMO, 5
Determine, with proof, all positive integers $k$ such that $$\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k$$ is an integer for every positive integer $n.$
1988 USAMO, 1
By a [i]pure repeating decimal[/i] (in base $10$), we mean a decimal $0.\overline{a_1\cdots a_k}$ which repeats in blocks of $k$ digits beginning at the decimal point. An example is $.243243243\cdots = \tfrac{9}{37}$. By a [i]mixed repeating decimal[/i] we mean a decimal $0.b_1\cdots b_m\overline{a_1\cdots a_k}$ which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is $.011363636\cdots = \tfrac{1}{88}$.
Prove that if a mixed repeating decimal is written as a fraction $\tfrac pq$ in lowest terms, then the denominator $q$ is divisible by $2$ or $5$ or both.
2021 USAMO, 5
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:
\begin{align*}
a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\
a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\
a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\
&\vdots & &\vdots \\
a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1}
\end{align*}
1984 USAMO, 5
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*}
P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}
Determine $n$.
2000 USAMO, 5
Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$
2010 USAMO, 1
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 $AB$.
2013 USAJMO, 6
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
2001 USAMO, 6
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.