Found problems: 85335
2018 AMC 12/AHSME, 2
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?
$\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$
JOM 2015 Shortlist, N6
Let $ p_i $ denote the $ i $-th prime number. Let $ n = \lfloor\alpha^{2015}\rfloor $, where $ \alpha $ is a positive real number such that $ 2 < \alpha < 2.7 $. Prove that $$ \displaystyle\sum_{2 \le p_i \le p_j \le n}\frac{1}{p_ip_j} < 2017 $$
1991 IMTS, 2
Show that every triangle can be dissected into nine convex nondegenrate pentagons.
2023 CCA Math Bonanza, I14
The decimal expansion of $37^9$ is $129A617B979C077$ for digits $A, B,$ and $C$. Find the three digit number $ABC$.
[i]Individual #14[/i]
2018 Taiwan TST Round 3, 1
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
2024 Harvard-MIT Mathematics Tournament, 8
Rishabh has $2024$ pairs of socks in a drawer. He draws socks from the drawer uniformly at random, without replacement, until he has drawn a pair of identical socks. Compute the expected number of unpaired socks he has drawn when he stops.
2015 Tournament of Towns, 1
[b](a)[/b] The integers $x$, $x^2$ and $x^3$ begin with the same digit. Does it imply that this digit is $1$?
[i]($2$ points) [/i]
[b](b)[/b] The same question for the integers $x, x^2, x^3, \cdots, x^{2015}$
[i]($3$ points)[/i]
.
Gheorghe Țițeica 2024, P3
Let $A,B\in\mathcal{M}_n(\mathbb{Z})$ and $p$ a prime number. Prove that $$\text{Tr}((A+B)^p)\equiv\text{Tr}(A^p+B^p)\pmod p.$$
2013 Canada National Olympiad, 2
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?
2020 Benelux, 3
Let $ABC$ be a triangle. The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$. Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.
2020 MBMT, 22
Find the product of all positive real solutions to the equation $x^{-x} + x^{\frac{1}{x}} = \frac{2021}{2020}.$
[i]Proposed by Gabriel Wu[/i]
1970 Dutch Mathematical Olympiad, 2
The equation $ x^3 - x^2 + ax - 2^n = 0$ has three integer roots. Determine $a$ and $n$.
2005 AMC 10, 20
An equiangular octagon has four sides of length $ 1$ and four sides of length $ \frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?
$ \textbf{(A)}\ \frac{7}{2}\qquad
\textbf{(B)}\ \frac{7\sqrt{2}}{2}\qquad
\textbf{(C)}\ \frac{5 \plus{} 4\sqrt{2}}{2}\qquad
\textbf{(D)}\ \frac{4 \plus{} 5\sqrt{2}}{2}\qquad
\textbf{(E)}\ 7$
2013 ELMO Shortlist, 3
In $\triangle ABC$, a point $D$ lies on line $BC$. The circumcircle of $ABD$ meets $AC$ at $F$ (other than $A$), and the circumcircle of $ADC$ meets $AB$ at $E$ (other than $A$). Prove that as $D$ varies, the circumcircle of $AEF$ always passes through a fixed point other than $A$, and that this point lies on the median from $A$ to $BC$.
[i]Proposed by Allen Liu[/i]
2014 BMT Spring, 15
Suppose a box contains $28$ balls: $1$ red, $2$ blue, $3$ yellow, $4$ orange, $5$ purple, $6$ green, and $7$ pink. One by one, each ball is removed uniformly at random and without replacement until all $28$ balls have been removed. Determine the probability that the most likely “scenario of exhaustion” occurs; that is, determine the probability that the first color to have all such balls removed from the box is red, that the second is blue, the third is yellow, the fourth is orange, the fifth is purple, the sixth is green, and the seventh is pink.
2020 Princeton University Math Competition, 5
Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A $3\times 4$ rectangle is cut into $n$ pieces by making straight line cuts. What is the minimum value of $n$ so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least $2021$?
2024 AMC 10, 9
Real numbers $a,b$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac$ and $bc$?
$
\textbf{(A) }-5 \qquad
\textbf{(B) }-\frac{10}{3} \qquad
\textbf{(C) }-\frac{10}{9} \qquad
\textbf{(D) }0 \qquad
\textbf{(E) }\frac{10}{9} \qquad
$
2019 Tuymaada Olympiad, 4
A quota of diplomas at the All-Russian Olympiad should be strictly less than $45\%$. More than $20$ students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than $45\%$. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to $45\%$. Prove that the number of participants should be at least doubled.
2021 CCA Math Bonanza, L4.2
Compute the number of (not necessarily convex) polygons in the coordinate plane with the following properties:
[list]
[*] If the coordinates of a vertex are $(x,y)$, then $x,y$ are integers and $1\leq |x|+|y|\leq 3$
[*] Every side of the polygon is parallel to either the x or y axis
[*] The point $(0,0)$ is contained in the interior of the polygon.
[/list]
[i]2021 CCA Math Bonanza Lightning Round #4.2[/i]
2024 IFYM, Sozopol, 8
Three piles of stones are given, initially containing 2000, 4000, and 4899 stones respectively. Ali and Baba play the following game, taking turns, with Ali starting first. In one move, a player can choose two piles and transfer some stones from one pile to the other, provided that at the end of the move, the pile from which the stones are moved has no fewer stones than the pile to which the stones are moved. The player who cannot make a move loses. Does either player have a winning strategy, and if so, who?
2001 AMC 8, 16
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
[asy]
draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8));
draw((2,-1)--(2,9),dashed);[/asy]
$ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ \frac{4}{5}\qquad\text{(E)}\ \frac{5}{6} $
Geometry Mathley 2011-12, 1.3
Let $ABC$ be an acute triangle with incenter $O$, orthocenter $H$, altitude $AD. AO$ meets $BC$ at $E$. Line through $D$ parallel to $OH$ meet $AB,AC$ at $M,N$, respectively. Let $I$ be the midpoint of $AE$, and $DI$ intersect $AB,AC$ at $P,Q$ respectively. $MQ$ meets $NP$ at $T$. Prove that $D,O, T$ are collinear.
Trần Quang Hùng
2006 Regional Competition For Advanced Students, 1
Let $ 0 < x <y$ be real numbers. Let
$ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$
be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.
2001 AIME Problems, 15
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.
May Olympiad L2 - geometry, 2014.2
In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.