Found problems: 85335
2023 BMT, 14
Right triangle $\vartriangle ABC$ with $\angle A = 30^o$ and $\angle B = 90^o$ is inscribed in a circle $\omega_1$ with radius $4$. Circle $\omega_2$ is drawn to be the largest circle outside of $\vartriangle ABC$ that is tangent to both $\overline{BC}$ and $\omega_1$, and circles $\omega_3$ and $\omega_4$ are drawn this same way for sides $\overline{AC}$ and $\overline{AB}$, respectively. Suppose that the intersection points of these smaller circles with the bigger circle are noted as points $D$, $E$, and $F$. Compute the area of triangle $\vartriangle DEF$.
2011 Princeton University Math Competition, A1 / B3
The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$.
2010 Saudi Arabia Pre-TST, 3.2
Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.
2015 Sharygin Geometry Olympiad, P24
The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$.
a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$.
Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$.
b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.
2011 AMC 12/AHSME, 19
At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$?
$ \textbf{(A)}\ 38\qquad
\textbf{(B)}\ 90 \qquad
\textbf{(C)}\ 154 \qquad
\textbf{(D)}\ 406 \qquad
\textbf{(E)}\ 1024$
2006 Mathematics for Its Sake, 2
Let be a natural number $ n. $ Solve in the set of $ 2\times 2 $ complex matrices the equation
$$ \begin{pmatrix} -2& 2007\\ 0&-2 \end{pmatrix} =X^{3n}-3X^n. $$
[i]Petru Vlad[/i]
1967 IMO Shortlist, 1
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.
2008 Tournament Of Towns, 2
Solve the system of equations $(n > 2)$
\[\begin{array}{c}\ \sqrt{x_1}+\sqrt{x_2+x_3+\cdots+x_n}=\sqrt{x_2}+\sqrt{x_3+x_4+\cdots+x_n+x_1}=\cdots=\sqrt{x_n}+\sqrt{x_1+x_2+\cdots+x_{n-1}} \end{array}, \] \[x_1-x_2=1.\]
2013 NIMO Problems, 2
In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$.
[i]Proposed by Eugene Chen[/i]
2021 Durer Math Competition Finals, 5
A torpedo set consists of $2$ pieces of $1 \times 4$, $4$ pieces of $1 \times 3$, $6$ pieces of $1 \times 2$ and $ 8$ pieces of $1 \times 1$ ships.
a) Can one put the whole set to a $10 \times 10$ table so that the ships do not even touch with corners? (The ships can be placed both horizontally and vertically.)
b) Can we solve this problem if we change $4$ pieces of $1 \times 1$ ships to $3$ pieces of $1 \times 2$ ships?
c) Can we solve the problem if we change the remaining $4$ pieces of $1 \times 1$ ships to one piece of $1 \times 3$ ship and one piece of $1 \times 2$ ship? (So the number of pieces are $2, 5, 10, 0$.)
1994 India Regional Mathematical Olympiad, 1
A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf?
1997 India National Olympiad, 1
Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]
2006 AMC 10, 18
A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?
$ \textbf{(A) } 10^4\cdot 26^2 \qquad \textbf{(B) } 10^3\cdot 26^3 \qquad \textbf{(C) } 5\cdot 10^4\cdot 26^2 \qquad \textbf{(D) } 10^2\cdot 26^4\\
\textbf{(E) } 5\cdot 10^3\cdot 26^3$
2025 Malaysian IMO Training Camp, 2
Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that:
1. Every positive integer appears in the sequence at least once, and;
2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$.
[i](Proposed by Ho Janson)[/i]
2004 Federal Competition For Advanced Students, Part 1, 2
A convex hexagon $ABCDEF$ with $AB = BC = a, CD = DE = b, EF = FA = c$ is inscribed in a circle. Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendicular diagonals.
2014 Taiwan TST Round 3, 1
Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]
2014 France Team Selection Test, 2
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
2016 Bulgaria National Olympiad, Problem 6
Let $n$ be positive integer.A square $A$ of side length $n$ is divided by $n^2$ unit squares. All unit squares are painted in $n$ distinct colors such that each color appears exactly $n$ times. Prove that there exists a positive integer $N$ , such that for any $n>N$ the following is true: There exists a square $B$ of side length $\sqrt{n}$ and side parallel to the sides of $A$ such that $B$ contains completely cells of $4$ distinct colors.
2005 Poland - Second Round, 2
In a convex quadrilateral $ABCD$, point $M$ is the midpoint of the diagonal $AC$. Prove that if $\angle BAD=\angle BMC=\angle CMD$, then a circle can be inscribed in quadrilateral $ABCD$.
1956 Putnam, A7
Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of $2.$
2023 CIIM, 3
Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ such that $A_{n+1} = f(A_n)$ for $n \geq 0$. If the matrix $A_0$ has no zero entries, determine the maximum number of indices $j \geq 0$ for which the matrix $A_j$ has any null entries.
2021 BMT, 4
Compute the sum of all real solutions to $4^x - 2021 \cdot 2^x + 1024 = 0$.
2015 Purple Comet Problems, 16
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made $60$ profit. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 profit. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars profit did Don make?
2005 Putnam, B5
Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that
(a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically)
and that
(b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$
Show that $P=0$ identically.
Durer Math Competition CD Finals - geometry, 2016.C2
Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.