Found problems: 85335
2020 AIME Problems, 4
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.
2009 Mathcenter Contest, 1
For any natural $n$ , define $n!!=(n!)!$ e.g. $3!!=(3!)!=6!=720$.
Let $a_1,a_2,...,a_n$ be a positive integer Prove that $$\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!}$$ is an integer.
[i](nooonuii)[/i]
2010 Today's Calculation Of Integral, 609
Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number.
1972 Tokyo University of Education entrance exam
2014 Poland - Second Round, 4.
$2n$ ($n\ge 2$) teams took part in the football league matches and there were $2n-1$ matchweeks. In each matchweek each team played one match. Any two teams met with each other during the matches in exactly one game. Moreover, in each match one team was the host and the second was a guest.
Say a team is [i] traveling[/i], if in any two consecutive matchweeks it was once a host and once a guest. Prove that there are at most two traveling teams.
2015 China Team Selection Test, 2
Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.
2022 BMT, 3
Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Katie’s score is strictly greater than Allie’s?
2020 Spain Mathematical Olympiad, 5
In an acute-angled triangle $ABC$, let $M$ be the midpoint of $AB$ and $P$ the foot of the altitude to $BC$. Prove that if $AC+BC = \sqrt{2}AB$, then the circumcircle of triangle $BMP$ is tangent to $AC$.
2021 Turkey Team Selection Test, 4
In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?
1983 Canada National Olympiad, 3
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?
2017 Iran Team Selection Test, 3
There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes.
What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]?
[i]Proposed by Amin Bahjati[/i]
2010 Regional Olympiad of Mexico Center Zone, 1
In the acute triangle $ABC$, $\angle BAC$ is less than $\angle ACB $. Let $AD$ be a diameter of $\omega$, the circle circumscribed to said triangle. Let $E$ be the point of intersection of the ray $AC$ and the tangent to $\omega$ passing through $B$. The perpendicular to $AD$ that passes through $E$ intersects the circle circumscribed to the triangle $BCE$, again, at the point $F$. Show that $CD$ is an angle bisector of $\angle BCF$.
2020 Purple Comet Problems, 9
Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1.
Find the product $abc$.
2016 CHMMC (Fall), 13
A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?
2025 AMC 8, 10
In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD = 3$ inches. Rectangle $ABCD$ is rotated $90^{\circ}$ clockwise about the midpoint of side $\overline{DC}$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?
[img]https://i.imgur.com/NyhZpL6.png[/img]
$\textbf{(A) }21 \qquad\textbf{(B) }22.25 \qquad\textbf{(C) }23\qquad\textbf{(D) }23.75 \qquad\textbf{(E) }25$
2014 Belarus Team Selection Test, 2
Prove that for all even positive integers $n$ the following inequality holds
a) $\{n\sqrt6\} > \frac{1}{n}$
b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $
(I. Voronovich)
1994 All-Russian Olympiad Regional Round, 10.5
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).
2018 Ramnicean Hope, 1
Show that $ 2/3+\sin 2018^{\circ } >0. $
[i]Costică Ambrinoc[/i]
2015 Dutch BxMO/EGMO TST, 1
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$.
Prove that $m$ is a divisor of $n$.
2018 Polish Junior MO First Round, 7
Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.
2010 JBMO Shortlist, 5
Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $
Prove that $xyz \leq 36$
2013 National Olympiad First Round, 6
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?
$
\textbf{(A)}\ 760
\qquad\textbf{(B)}\ 756
\qquad\textbf{(C)}\ 755
\qquad\textbf{(D)}\ 752
\qquad\textbf{(E)}\ 750
$
2004 Turkey Junior National Olympiad, 3
On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school?
2022 VIASM Summer Challenge, Problem 1
Given prime numbers $p$ and $q.$
a) Assume that $2^xq=p^y+1,$ with $x,y$ are integers greater than $1$. Can $x$ be a composite number?
b) Assume that $2^uq=p^v-1,$ with $u$ is a prime number and $v$ is an integer greater than $1$. Find all possible values of $p.$
1968 Yugoslav Team Selection Test, Problem 1
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
2009 Mexico National Olympiad, 2
Consider a triangle $ABC$ and a point $M$ on side $BC$. Let $P$ be the intersection of the perpendiculars from $M$ to $AB$ and from $B$ to $BC$, and let $Q$ be the intersection of the perpendiculars from $M$ to $AC$ and from $C$ to $BC$. Show that $PQ$ is perpendicular to $AM$ if and only if $M$ is the midpoint of $BC$.