Found problems: 85335
ICMC 8, 3
Let $V$ be a subspace of the vector space $\mathbb{R}^{2 \times 2}$ of $2$-by-$2$ real matrices. We call $V$ nice if for any linearly independent $A, B \in V$, $AB \neq BA$. Find the maximum dimension of a nice subspace of $\mathbb{R}^{2 \times 2}$.
1967 IMO Shortlist, 4
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.
2020 Harvard-MIT Mathematics Tournament, 8
Let $P(x)$ be the unique polynomial of degree at most $2020$ satisfying $P(k^2)=k$ for $k=0,1,2,\dots,2020$. Compute $P(2021^2)$.
[i]Proposed by Milan Haiman.[/i]
1990 AMC 12/AHSME, 4
Let $ABCD$ be a parallelogram with $\angle ABC=120^\circ$, $AB=16$ and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to
$\textbf{(A) }1\qquad
\textbf{(B) }2\qquad
\textbf{(C) }3\qquad
\textbf{(D) }4\qquad
\textbf{(E) }5$
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(16,0),C=(26,10*sqrt(3)),D=(10,10*sqrt(3)),E=(0,10*sqrt(3));
draw(A--B--C--E--B--A--D);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,N);
label("$E$",E,N);
label("$F$",extension(A,D,B,E),W);
label("$4$",(D+E)/2,N);
label("$16$",(8,0),S);
label("$10$",(B+C)/2,SE);
[/asy]
2011 National Olympiad First Round, 15
For which pair $(a,b)$, there is no positive real pair $(x,y)$ satisfying $x+2y < a$ and $xy > b$ ?
$\textbf{(A)}\ \left (\frac{15}{7}, \frac{4}{7}\right ) \qquad\textbf{(B)}\ \left (\frac{18}{11}, \frac{1}{3}\right ) \qquad\textbf{(C)}\ \left (\frac{5}{7}, \frac{1}{16}\right ) \qquad\textbf{(D)}\ \left (\frac{6}{7}, \frac{1}{11}\right ) \qquad\textbf{(E)}\ \text{None}$
2010 Irish Math Olympiad, 1
There are $14$ boys in a class. Each boy is asked how many other boys in the class have his first name, and how many have his last name. It turns out that each number from $0$ to $6$ occurs among the answers.
Prove that there are two boys in the class with the same first name and the same last name.
1968 AMC 12/AHSME, 13
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m\ne0$, $n\ne0$, then the sum of the roots is:
$\textbf{(A)}\ -\dfrac{1}{2} \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ \dfrac{1}{2} \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \text{Undetermined} $
2024 Kyiv City MO Round 2, Problem 3
Let $AH_A, BH_B, CH_C$ be the altitudes of the triangle $ABC$. Points $A_1$ and $C_1$ are the projections of the point $H_B$ onto the sides $AB$ and $BC$, respectively. $B_1$ is the projection of $B$ onto $H_AH_C$. Prove that the diameter of the circumscribed circle of $\triangle A_1B_1C_1$ is equal to $BH_B$.
[i]Proposed by Anton Trygub[/i]
2017 May Olympiad, 5
Ababa plays with a word made up of the letters of his name and has set certain rules:
If you find an $A$ followed immediately by a $B$, you can substitute $BAA$ for them.
If you find two consecutive $B$'s, you can delete them.
If you find three consecutive $A$'s, you can delete them.
Ababa begins with the word $ABABABAABAAB$.
With the above rules, how many letters do you have the shortest word you can come up with?
Why can't you come up with one more word shorter?
2021 Brazil Team Selection Test, 3
Let $ABC$ be an acute triangle with $AC>CB$ and let $M$ be the midpoint of side $AB$. Denote by $Q$ the midpoint of the big arc $AB$ which cointais $C$ and by $B_1$ the point inside $AC$ such that $BC=CB_1$. $B_1Q$ touches $BC$ in $E$ and $K$ is the intersection of $(BB_1M)$ and $(ABC)$. Prove that $KC$ bissects $B_1E$.
1990 IMO Longlists, 60
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$
[i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers.
[i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$
2024 District Olympiad, P4
Let $H{}$ be the orthocenter of the triangle $ABC{}$ and $X{}$ be the midpoint of the side $BC.$ The perpendicular at $H{}$ to $HX{}$ intersects the sides $(AB)$ and $(AC)$ at $Y{}$ and $Z{}$ respectively. Let $O{}$ be the circumcenter of $ABC{}$ and $O'$ be the circumcenter of $BHC.$ [list=a]
[*]Prove that $HY=HZ.$
[*]Prove that $\overrightarrow{AY}+\overrightarrow{AZ}=2\overrightarrow{OO'}.$
[/list]
2018 Federal Competition For Advanced Students, P2, 6
Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$.
[i](Proposed by Walther Janous)[/i]
2017 Saudi Arabia BMO TST, 4
Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$.
a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number.
b) A positive integer is called [i]Fib-unique[/i] if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.
1988 IMO Longlists, 55
Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]
OIFMAT II 2012, 3
In the interior of an equilateral triangle $ ABC $ a point $ P $ is chosen such that $ PA ^2 = PB ^2 + PC ^2 $. Find the measure of $ \angle BPC $.
1968 IMO, 2
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
2017 MIG, 6
Thomas is to read $225$ pages of a book over the summer. He decides to read one page the first day, three pages the second day, five pages the third day, and so on, each day reading two more pages than the previous. How many days will it take Thomas to finish reading the book?
$\textbf{(A) } 11\qquad\textbf{(B) } 12\qquad\textbf{(C) } 13\qquad\textbf{(D) } 14\qquad\textbf{(E) } 15$
2013 Romanian Master of Mathematics, 6
A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
2023 CMIMC Geometry, 6
Let $ABCD$ be a regular tetrahedron. Suppose points $X$, $Y$, and $Z$ lie on rays $AB$, $AC$, and $AD$ respectively such that $XY=YZ=7$ and $XZ=5$. Moreover, the lengths $AX$, $AY$, and $AZ$ are all distinct. Find the volume of tetrahedron $AXYZ$.
[i]Proposed by Connor Gordon[/i]
2021 Miklós Schweitzer, 7
If the binary representations of the positive integers $k$ and $n$ are $k = \sum_{i=0}^{\infty} k_i 2^i$ and $n = \sum_{i=0}^{\infty} n_i 2^i$, then the logical sum of these numbers is
\[ k \oplus n =\sum_{i=0}^{\infty} |k_i-n_i|2^i. \]
Let $N$ be an arbitrary positive integer and $(c_k)_{k \in \mathbb{N}}$ be a sequence of complex numbers such that for all $k \in \mathbb{N}$, $ |c_k| \le 1$. Prove that there exist positive constants $C$ and $\delta$ such that
\[ \int_{[-\pi,\pi] \times [-\pi, \pi]} \sup_{n<N, n \in \mathbb{N}} \frac{1}{N} \Big| \sum_{k=1}^{n} c_k e^{i(kx+(k \oplus n) y)} \Big| \mathrm d(x,y) \le C \cdot N^{-\delta} \]
holds.
2015 Baltic Way, 16
Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]
2019 District Olympiad, 3
Let $G$ be a finite group and let $x_1,…,x_n$ be an enumeration of its elements. We consider the matrix $(a_{ij})_{1 \le i,j \le n},$ where $a_{ij}=0$ if $x_ix_j^{-1}=x_jx_i^{-1},$ and $a_{ij}=1$ otherwise. Find the parity of the integer $\det(a_{ij}).$
2022 Sharygin Geometry Olympiad, 9.6
Lateral sidelines $AB$ and $CD$ of a trapezoid $ABCD$ ($AD >BC$) meet at point $P$. Let $Q$ be a point of segment $AD$ such that $BQ = CQ$. Prove that the line passing through the circumcenters of triangles $AQC$ and $BQD$ is perpendicular to $PQ$.
2019 PUMaC Geometry B, 8
Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$