Found problems: 85335
2007 Princeton University Math Competition, 8
Find the biggest $ n < 2007 $ such that there exists a partition of the integers from $1$ to $n$ into two sets the sums of the squares of whose elements are equal.
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
2016 Romanian Master of Mathematics Shortlist, A2
Let $p > 3$ be a prime number, and let $F_p$ denote the (fi nite) set of residue classes modulo $p$.
Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$.
[i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$
[/i] appearing in $P$.
2025 AIME, 10
The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.
[asy]
unitsize(20);
add(grid(9,3));
draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2));
draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));
real a = 0.5;
label("5",(a,a));
label("6",(1+a,a));
label("1",(2+a,a));
label("8",(3+a,a));
label("4",(4+a,a));
label("7",(5+a,a));
label("9",(6+a,a));
label("2",(7+a,a));
label("3",(8+a,a));
label("3",(a,1+a));
label("7",(1+a,1+a));
label("9",(2+a,1+a));
label("5",(3+a,1+a));
label("2",(4+a,1+a));
label("1",(5+a,1+a));
label("6",(6+a,1+a));
label("8",(7+a,1+a));
label("4",(8+a,1+a));
label("4",(a,2+a));
label("2",(1+a,2+a));
label("8",(2+a,2+a));
label("9",(3+a,2+a));
label("6",(4+a,2+a));
label("3",(5+a,2+a));
label("1",(6+a,2+a));
label("7",(7+a,2+a));
label("5",(8+a,2+a));
[/asy]
The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.
2007 Indonesia TST, 3
Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.
1960 Putnam, A7
Let $N(n)$ denote the smallest positive integer $N$ such that $x^N =e$ for every element $x$ of the symmetric group $S_n$, where $e$ denotes the identity permutation. Prove that if $n>1,$
$$\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\
1\; \text{otherwise}.
\end{cases}$$
Geometry Mathley 2011-12, 4.1
Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$
Bùi Quang Tuấn
2015 Purple Comet Problems, 18
You have many identical cube-shaped wooden blocks. You have four colors of paint to use, and you paint
each face of each block a solid color so that each block has at least one face painted with each of the four
colors. Find the number of distinguishable ways you could paint the blocks. (Two blocks are
distinguishable if you cannot rotate one block so that it looks identical to the other block.)
2012 ELMO Shortlist, 7
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$.
[i]Alex Zhu.[/i]
2019 MIG, 5
How many distinct prime factors does the number $36$ have?
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }9\qquad\textbf{(E) }15$
2022 Math Prize for Girls Problems, 8
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?
2001 China Team Selection Test, 2
$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?
2010 Contests, 2
There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.
2018 Korea Winter Program Practice Test, 1
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.
2019 Brazil Undergrad MO, 1
Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$
with rational entries and size $ 2019 $ such that:
a) $ A ^ 3 + 6A ^ 2-2I = 0 $?
b) $ A ^ 4 + 6A ^ 3-2I = 0 $?
2005 Today's Calculation Of Integral, 90
Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$
where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
2022 MIG, 1
What is $4^0 - 3^1 - 2^2 - 1^3$?
$\textbf{(A) }{-}8\qquad\textbf{(B) }{-}7\qquad\textbf{(C) }{-}5\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2021 Latvia Baltic Way TST, P16
A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$.
2016 APMC, 5
Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$
Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$
Kyiv City MO Juniors 2003+ geometry, 2020.9.4
Let the point $D$ lie on the arc $AC$ of the circumcircle of the triangle $ABC$ ($AB < BC$), which does not contain the point $B$. On the side $AC$ are selected an arbitrary point $X$ and a point $X'$ for which $\angle ABX= \angle CBX'$. Prove that regardless of the choice of the point $X$, the circle circumscribed around $\vartriangle DXX'$, passes through a fixed point, which is different from point $D$.
(Nikolaev Arseniy)
2008 Singapore Junior Math Olympiad, 3
In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.
2016 India Regional Mathematical Olympiad, 6
A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\dots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.
1976 Putnam, 5
Evaluate $$\sum_{k=0}^n (-1)^k \binom{n}{k} (x-k)^n.$$
2015 AIME Problems, 9
A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.
[asy]
import three; import solids;
size(5cm);
currentprojection=orthographic(1,-1/6,1/6);
draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight);
triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2));
draw(X--X+A--X+A+B--X+A+B+C);
draw(X--X+B--X+A+B);
draw(X--X+C--X+A+C--X+A+B+C);
draw(X+A--X+A+C);
draw(X+C--X+C+B--X+A+B+C,linetype("2 4"));
draw(X+B--X+C+B,linetype("2 4"));
draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight);
draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0));
draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0));
draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4"));
[/asy]
2022 CMIMC, 1.5
At CMIMC headquarters, there is a row of $n$ lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things:
[list]
[*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken).
[*] Check if a given lightbulb is on or off.
[*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on.
[/list]
Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most $n$ seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value $n$ such that he can complete his task on time?
[i]Proposed by Adam Bertelli[/i]