Found problems: 85335
1988 AIME Problems, 14
Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form
\[ 12x^2 + bxy + cy^2 + d = 0. \]
Find the product $bc$.
1999 Federal Competition For Advanced Students, Part 2, 1
Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.
2018 Latvia Baltic Way TST, P8
Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:
[list]
[*] for every pair of students there is exactly one problem that was solved by both students;
[*] for every pair of problems there is exactly one student who solved both of them;
[*] one specific problem was solved by Laura and exactly $n$ other students.
[/list]
Determine the number of students in Laura's class.
2003 Kazakhstan National Olympiad, 5
Prove that for all primes $p>3$, $\binom{2p}{p}-2$ is divisible by $p^3$
2005 Junior Balkan Team Selection Tests - Romania, 5
On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$.
Prove that the triangles $QCP$ and $MCN$ have the same area.
1978 IMO Longlists, 54
Let $p, q$ and $r$ be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes $\alpha, \beta$, and $\gamma$, containing $p, q$, and $r$ respectively, that are perpendicular to each other $(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).$
1978 IMO Longlists, 7
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
1993 IMO Shortlist, 3
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
2007 Stars of Mathematics, 4
Show that any subset of $ A=\{ 1,2,...,2007\} $ having $ 27 $ elements contains three distinct numbers such that the greatest common divisor of two of them divides the other one.
[i]Dan Schwarz[/i]
1997 Tournament Of Towns, (564) 5
Dima invented a secret code in which every letter is replaced by a word no longer than $10$ letters. A code is called “good” if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima’s code, every possible word of at most $10000$ letters can be decoded in only one way. Does it follow that Dima’s code is good? (Note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has $ 33$ letters! A word is any sequence of letters.)
(D Piontkovskiy, S Shalunov)
2019 AIME Problems, 3
In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.
2023 Math Prize for Girls Problems, 1
The frame of a painting has the form of a $105^{\prime\prime}$ by $105^{\prime\prime}$ square with a $95^{\prime\prime}$ by $95^{\prime\prime}$ square removed from its center. The frame is built out of congruent isosceles trapezoids with angles measuring $45^\circ$ and $135^\circ$. Each trapezoid has one base on the frame's outer edge and one base on the frame's inner edge. Each outer edge of the frame contains an odd number of trapezoid bases that alternate long, short, long, short, etc. What is the maximum possible number of trapezoids in the frame?
I Soros Olympiad 1994-95 (Rus + Ukr), 9.6
In the triangle $ABC$, the orthocenter $H$ lies on the inscribed circle. Is this triangle necessarily isosceles?
2015 AMC 10, 22
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
$\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$
2011 Argentina Team Selection Test, 3
Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$.
Show that $SO \perp XY$.
2016 Indonesia TST, 4
In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.
1962 Czech and Slovak Olympiad III A, 4
Consider a circle $k$ with center $S$ and radius $r$. Let a point $A\neq S$ be given with $SA=d<r$. Consider a light ray emitted at point $A$, reflected at point $B\in k$, further reflected in point $C\in k$, which then passes through the original point $A$. Compute the sinus of convex angle $SAB$ in terms of $d,r$ and discuss conditions of solvability.
2000 AMC 10, 14
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered?
$\text{(A)}\ 71 \qquad\text{(B)}\ 76 \qquad\text{(C)}\ 80 \qquad\text{(D)}\ 82 \qquad\text{(E)}\ 91$
2018 BMT Spring, 9
Compute the following: $$\sum^{99}_{x=0} (x^2 + 1)^{-1} \,\,\, (mod \,\,\,199)$$ where $x^{-1}$ is the value $0 \le y \le 199$ such that $xy - 1$ is divisible by $199$.
2002 IMO Shortlist, 1
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
2015 Iran MO (2nd Round), 1
Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.
2019 BMT Spring, 4
There exists one pair of positive integers $ a, b $ such that $ 100 > a > b > 0 $ and $ \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{2}{35} $. Find $ a + b $.
2006 QEDMO 2nd, 4
Let $ABCD$ be a cyclic quadrilateral. Let $X$ be the foot of the perpendicular from the point $A$ to the line $BC$, let $Y$ be the foot of the perpendicular from the point $B$ to the line $AC$, let $Z$ be the foot of the perpendicular from the point $A$ to the line $CD$, let $W$ be the foot of the perpendicular from the point $D$ to the line $AC$.
Prove that $XY\parallel ZW$.
Darij
2022 MIG, 9
A circle with area $\tfrac{36}{\pi}$ has the same perimeter as a square with what side length?
$\textbf{(A) }\frac{9}{\pi}\qquad\textbf{(B) }3\qquad\textbf{(C) }\pi\qquad\textbf{(D) }6\qquad\textbf{(E) }\pi^2$
1990 IberoAmerican, 1
Let $f$ be a function defined for the non-negative integers, such that:
a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$.
b) $f(n+1)=f(n)-1$ otherwise.
i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$.
ii) Find $f(2^{1990})$.