Found problems: 85335
2020 Thailand TST, 2
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
(Hungary)
2011 Laurențiu Duican, 1
Let be three positive real numbers $ x,y,z. $ Prove that there is a group of real numbers that contain the elements $ x+y/z $ and $ x+z/y $ and in which these two elements are inverses to each other.
[i]D.M. Bătinețu[/i]
2012 NIMO Problems, 1
Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$.
[i]Proposed by Eugene Chen[/i]
1957 Moscow Mathematical Olympiad, 363
Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.
2024 Oral Moscow Geometry Olympiad, 2
Petya drew a pentagon $ABCDE$ on the plane. After that, Vasya marked all the points $S$ in a given half-space relative to the plane of the pentagon so that in the pyramid $SABCD$ exactly two side faces are perpendicular to the plane of the base $ABCD$, and the height is $1$. How many points could have Vasya?
1999 Junior Balkan Team Selection Tests - Romania, 4
Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $
If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then
$$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$
[i]Vasile Pop[/i]
2020 Yasinsky Geometry Olympiad, 5
Let $AL$ be the bisector of triangle $ABC$. Circle $\omega_1$ is circumscribed around triangle $ABL$. Tangent to $\omega_1$ at point $B$ intersects the extension of $AL$ at point $K$. The circle $\omega_2$ circumscribed around the triangle $CKL$ intersects $\omega_1$ a second time at point $Q$, with $Q$ lying on the side $AC$. Find the value of the angle $ABC$.
(Vladislav Radomsky)
2012 Online Math Open Problems, 23
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Author: Ray Li[/i]
1990 IMO Shortlist, 8
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$
1997 Poland - Second Round, 1
For the real number $a$ find the number of solutions $(x, y, z)$ of a system of the equations:
$\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.$
2013 Hanoi Open Mathematics Competitions, 1
How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square?
(A): $1$, (B): $2$, (C): $4$, (D): $8$, (E) None of the above.
2017 HMIC, 5
Let $S$ be the set $\{-1, 1\}^n$, that is, $n$-tuples such that each coordinate is either $-1$ or $1$. For \[s = (s_1, s_2, \ldots, s_n), t = (t_1, t_2, \ldots, t_n) \in \{-1, 1\}^n,\] define $s \odot t = (s_1t_1, s_2t_2, \ldots, s_nt_n)$.
Let $c$ be a positive constant, let $f : S \to \{-1, 1\}$ be a function such that there are at least $(1-c) \cdot 2^{2n}$ pairs $(s, t)$ with $s, t \in S$ such that $f(s \odot t) = f(s)f(t)$. Show that there exists a function $f'$ such that $f'(s \odot t) = f'(s)f'(t)$ for all $s, t \in S$ and $f(s) = f'(s)$ for at least $(1-10c) \cdot 2^n$ values of $s \in S$.
2019 BMT Spring, 14
A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$?
You need not simplify/compute exponents.
2018 CMIMC Individual Finals, 1
How many nonnegative integers with at most $40$ digits consisting of entirely zeroes and ones are divisible by $11$?
1989 India National Olympiad, 6
Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.
2022 ISI Entrance Examination, 1
Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by $0$ or $1$ .
[list=a]
[*] In how many ways can this be done such that each row sum and each column sum is even?
[*] In how many ways can this be done such that each row sum and each column sum is odd?
[/list]
2001 Stanford Mathematics Tournament, 4
For what values of $a$ does the system of equations
\[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?
2020 BAMO, E/3
The integer $202020$ is a multiple of $91$. For each positive integer $n$, show how $n$ additional $2$'s may be inserted into the digits of $202020$ such that the resulting $(n+6)$-digit number is also a multiple of $91$.
For example, a possible way to do this when $n=5$ is [u]2[/u]2020[u]2[/u]20[u]222[/u] (the inserted $2$'s are underlined).
2023 Romanian Master of Mathematics Shortlist, G2
Let $ABCD$ be a cyclic quadrilateral. Let $DA$ and $BC$ intersect at $E$ and let $AB$ and $CD$
intersect at $F$. Assume that $A, E, F$ all lie on the same side of $BD$. Let $P$ be on segment $DA$
such that $\angle CPD = \angle CBP$, and let $Q$ be on segment $CD$ such that $\angle DQA = \angle QBA$. Let $AC$ and $PQ$ meet at $X$. Prove that, if $EX = EP$, then $EF$ is perpendicular to $AC$.
2006 AMC 8, 18
A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
$ \textbf{(A)}\ \dfrac{1}{9} \qquad
\textbf{(B)}\ \dfrac{1}{4} \qquad
\textbf{(C)}\ \dfrac{4}{9} \qquad
\textbf{(D)}\ \dfrac{5}{9} \qquad
\textbf{(E)}\ \dfrac{19}{27}$
2009 Junior Balkan Team Selection Tests - Romania, 4
Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that:
$$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$
2023 Harvard-MIT Mathematics Tournament, 2
Points $X$, $Y$, and $Z$ lie on a circle with center $O$ such that $XY=12$. Points $A$ and $B$ lie on segment $XY$ such that $OA=AZ=ZB=BO=5$. Compute $AB$.
2013 Stars Of Mathematics, 2
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
[i](Dan Schwarz)[/i]
1993 Tournament Of Towns, (390) 2
Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees.
(Folklore)
2018 BMT Spring, 3
Find the minimal $N$ such that any $N$-element subset of $\{1, 2, 3, 4,...,7\}$ has a subset $S$ such that the sum of elements of $S$ is divisible by $7$.