Found problems: 85335
2020 AMC 12/AHSME, 8
How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$
2016 Kosovo Team Selection Test, 4
It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied
$f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$
If $g(x)=f(x)-x+1$ then find $g(2016)$ .
2019 HMNT, 8
Omkar, Krit1, Krit2, and Krit3 are sharing $x > 0$ pints of soup for dinner. Omkar always takes $1$ pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit1 always takes $\frac16$ of what is left, Krit2 always takes $\frac15$ of what is left, and Krit3 always takes $\frac14$ of what is left. They take soup in the order of Omkar, Krit1, Krit2, Krit3, and then cycle through this order until no soup remains. Find all $x$ for which everyone gets the same amount of soup.
2019 SG Originals, Q5
In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes.
[i]Proposed by DVDthe1st, mzy and jjax[/i]
2016 Switzerland Team Selection Test, Problem 2
Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$
1999 AMC 8, 20
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.
Which of the following is the front view for the stack map in Fig. 4?
[asy]
unitsize(24);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((0,1)--(2,1));
draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle);
draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3));
draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3));
draw((17/3,7/3)--(14/3,7/3));
draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0));
draw((5,1)--(6,1)--(6,0));
draw((20/3,4/3)--(6,4/3));
draw((17/3,13/3)--(16/3,14/3));
draw((17/3,10/3)--(16/3,11/3));
draw((14/3,10/3)--(13/3,11/3));
draw((5,2)--(13/3,8/3));
draw((5,1)--(13/3,5/3));
draw((6,2)--(17/3,7/3));
draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle);
draw((11,3)--(10,3)--(10,0));
draw((11,2)--(9,2));
draw((11,1)--(9,1));
draw((13,0)--(16,0)--(16,2)--(13,2)--cycle);
draw((13,1)--(16,1));
draw((14,0)--(14,2));
draw((15,0)--(15,2));
label("Figure 1",(1,0),S);
label("Figure 2",(17/3,0),S);
label("Figure 3",(10,0),S);
label("Figure 4",(14.5,0),S);
label("$1$",(1.5,.2),N);
label("$2$",(.5,.2),N);
label("$3$",(.5,1.2),N);
label("$4$",(1.5,1.2),N);
label("$1$",(13.5,.2),N);
label("$3$",(14.5,.2),N);
label("$1$",(15.5,.2),N);
label("$2$",(13.5,1.2),N);
label("$2$",(14.5,1.2),N);
label("$4$",(15.5,1.2),N);[/asy]
[asy]
unitsize(18);
draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle);
draw((0,3)--(1,3));
draw((0,2)--(1,2)--(1,0));
draw((0,1)--(3,1));
draw((2,0)--(2,2));
draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle);
draw((8,3)--(7,3)--(7,0));
draw((8,2)--(6,2)--(6,0));
draw((8,1)--(5,1));
draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle);
draw((12,3)--(11,3)--(11,0));
draw((12,2)--(10,2));
draw((12,1)--(10,1));
draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle);
draw((17,3)--(16,3));
draw((17,2)--(16,2)--(16,0));
draw((17,1)--(14,1));
draw((15,0)--(15,2));
draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle);
draw((22,3)--(20,3));
draw((22,2)--(20,2));
draw((22,1)--(20,1)--(20,0));
draw((21,0)--(21,4));
label("(A)",(1.5,0),S);
label("(B)",(6.5,0),S);
label("(C)",(11,0),S);
label("(D)",(15.5,0),S);
label("(E)",(20.5,0),S);[/asy]
2013 Brazil Team Selection Test, 1
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.
[i]Proposed by Warut Suksompong, Thailand[/i]
2019 Latvia Baltic Way TST, 7
Two sequences $b_i$, $c_i$, $0 \le i \le 100$ contain positive integers, except $c_0=0$ and $b_{100}=0$.
Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely $1$ km long. Towns, which are connected by a road or a sequence of roads, are called [i]neighbours[/i]. The length of the shortest path between two towns $X$ and $Y$ is denoted as [i]distance[/i]. It is known that the greatest [i]distance[/i] between two towns in Graphland is $100$ km. Also the following property holds for every pair $X$ and $Y$ of towns (not necessarily distinct): if the [i]distance[/i] between $X$ and $Y$ is exactly $k$ km, then $Y$ has exactly $b_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k+1$ from $X$, and exactly $c_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k-1$ from $X$.
Prove that $$\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}$$ is a positive integer.
2001 IMO Shortlist, 1
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
2009 AMC 12/AHSME, 6
Suppose that $ P\equal{}2^m$ and $ Q\equal{}3^n$. Which of the following is equal to $ 12^{mn}$ for every pair of integers $ (m,n)$?
$ \textbf{(A)}\ P^2Q \qquad
\textbf{(B)}\ P^nQ^m \qquad
\textbf{(C)}\ P^nQ^{2m} \qquad
\textbf{(D)}\ P^{2m}Q^n \qquad
\textbf{(E)}\ P^{2n}Q^m$
2021 Malaysia IMONST 1, Primary
International Mathematical Olympiad National Selection Test
Malaysia 2021 Round 1 Primary
Time: 2.5 hours [hide=Rules]
$\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
$\bullet$ No mark is deducted for a wrong answer.
$\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide]
[b]Part A[/b] (1 point each)
p1. Faris has six cubes on his table. The cubes have a total volume of $2021$ cm$^3$. Five of the cubes have side lengths $5$ cm, $5$ cm, $6$ cm, $6$ cm, and $11$ cm. What is the side length of the sixth cube (in cm)?
p2. What is the sum of the first $200$ even positive integers?
p3. Anushri writes down five positive integers on a paper. The numbers are all different, and are all smaller than $10$. If we add any two of the numbers on the paper, then the result is never $10$. What is the number that Anushri writes down for certain?
p4. If the time now is $10.00$ AM, what is the time $1,000$ hours from now? Note: Enter the answer in a $12$-hour system, without minutes and AM/PM. For example, if the answer is $9.00$ PM, just enter $9$.
p5. Aminah owns a car worth $10,000$ RM. She sells it to Neesha at a $10\%$ profit. Neesha sells the car back to Aminah at a $10\%$ loss. How much money did Aminah make from the two transactions, in RM?
[b]Part B[/b] (2 points each)
p6. Alvin takes 250 small cubes of side length $1$ cm and glues them together to make a cuboid of size $5$ cm $\times 5$ cm $\times 10$ cm. He paints all the faces of the large cuboid with the color green. How many of the small cubes are painted by Alvin?
p7. Cikgu Emma and Cikgu Tan select one integer each (the integers do not have to be positive). The product of the two integers they selected is $2021$. How many possible integers could have been selected by Cikgu Emma?
p8. A three-digit number is called [i]superb[/i] if the first digit is equal to the sum of the other two digits. For example, $431$ and $909$ are superb numbers. How many superb numbers are there?
p9. Given positive integers $a, b, c$, and $d$ that satisfy the equation $4a = 5b =6c = 7d$. What is the smallest possible value of $ b$?
p10. Find the smallest positive integer n such that the digit sum of n is divisible by $5$, and the digit sum of $n + 1$ is also divisible by $5$.
Note: The digit sum of $1440$ is $1 + 4 + 4 + 0 = 9$.
[b]Part C[/b] (3 points each)
p11. Adam draws $7$ circles on a paper, with radii $ 1$ cm, $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm, and $7$ cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm$^2$) between the total area of the red circles and the total area of the blue circles?
p12. The number $2021$ has a special property that the sum of any two neighboring digits in the number is a prime number ($2 + 0 = 2$, $0 + 2 = 2$, and $2 + 1 = 3$ are all prime numbers). Among numbers from $2021$ to $2041$, how many of them have this property?
p13. Clarissa opens a pet shop that sells three types of pets: gold shes, hamsters, and parrots. The pets inside the shop together have a total of $14$ wings, $24$ heads, and $62$ legs. How many gold shes are there inside Clarissa's shop?
p14. A positive integer $n$ is called [i]special [/i] if $n$ is divisible by $4$, $n+1$ is divisible by $5$, and $n + 2$ is divisible by $6$. How many special integers smaller than $1000$ are there?
p15. Suppose that this decade begins on $ 1$ January $2020$ (which is a Wednesday) and the next decade begins on $ 1$ January $2030$. How many Wednesdays are there in this decade?
[b]Part D[/b] (4 points each)
p16. Given an isosceles triangle $ABC$ with $AB = AC$. Let D be a point on $AB$ such that $CD$ is the bisector of $\angle ACB$. If $CB = CD$, what is $\angle ADC$, in degrees?
p17. Determine the number of isosceles triangles with the following properties:
all the sides have integer lengths (in cm), and the longest side has length $21$ cm.
p18. Ha z marks $k$ points on the circumference of a circle. He connects every point to every other point with straight lines. If there are $210$ lines formed, what is $k$?
p19. What is the smallest positive multiple of $24$ that can be written using digits $4$ and $5$ only?
p20. In a mathematical competition, there are $2021$ participants. Gold, silver, and bronze medals are awarded to the winners as follows:
(i) the number of silver medals is at least twice the number of gold medals,
(ii) the number of bronze medals is at least twice the number of silver medals,
(iii) the number of all medals is not more than $40\%$ of the number of participants.
The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded?
PS. Problems 11-20 were also used in [url=https://artofproblemsolving.com/community/c4h2676837p23203256]Juniors [/url]as 1-10.
2019 PUMaC Team Round, 2
In a standard game of Rock–Paper–Scissors, two players repeatedly choose between rock, paper, and scissors, until they choose different options. Rock beats scissors, scissors beats paper, and paper beats rock. Nathan knows that on each turn, Richard randomly chooses paper with probability $33\%$, scissors with probability $44\%$, and rock with probability $23\%$. If Nathan plays optimally against Richard, the probability that Nathan wins is expressible as $a/b$ where $a$ and $b$ are coprime positive integers. Find $a + b$.
2019 Korea - Final Round, 5
Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).
2018 VJIMC, 4
Determine all possible (finite or infinite) values of
\[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\]
if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying
\[f(f(x))^4-f(f(x))+f(x)=1\]
for all $x \in \mathbb{R}$.
2017 BMT Spring, 10
You and your friend play a game on a $ 7 \times 7$ grid of buckets. Your friend chooses $5$ “lucky” buckets by marking an “$X$” on the bottom that you cannot see. However, he tells you that they either form a vertical, or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets:
either $\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\}$,
or $\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, a)\}$,
with $1\le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a “lucky” bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is “lucky”?
2009 Postal Coaching, 3
Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions
(i) $f(0, 0) = 1$, $f(0, 1) = 1$
(ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and
(iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of
$$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$
1952 AMC 12/AHSME, 41
Increasing the radius of a cylinder by $ 6$ units increased the volume by $ y$ cubic units. Increasing the altitude of the cylinder by $ 6$ units also increases the volume by $ y$ cubic units. If the original altitude is $ 2$, then the original radius is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 6\pi \qquad\textbf{(E)}\ 8$
1967 IMO Longlists, 20
In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
2016 BMT Spring, 6
How many integers less than $400$ have exactly $3$ factors that are perfect squares?
2003 Rioplatense Mathematical Olympiad, Level 3, 2
Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?
2019 HMNT, 6
Find all ordered pairs $(a,b)$ of positive integers such that $2a + 1$ divides $3b - 1$ and $2b + 1$ divides $3a -1$.
2023 USAMTS Problems, 3
Lizzie and Alex are playing a game on the whiteboard. Initially, $n$ twos are written
on the board. On a player’s turn they must either
1. change any single positive number to 0, or
2. subtract one from any positive number of positive numbers on the board.
The game ends once all numbers are 0, and the last player who made a move wins. If Lizzie
always plays first, find all $n$ for which Lizzie has a winning strategy.
2013 NIMO Problems, 4
Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey
\begin{align*}
0.1 &= \sin \gamma \cos \theta \sin \alpha, \\
0.2 &= \sin \gamma \sin \theta \cos \alpha, \\
0.3 &= \cos \gamma \cos \theta \sin \beta, \\
0.4 &= \cos \gamma \sin \theta \cos \beta, \\
0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert.
\end{align*}[i]Proposed by Evan Chen[/i]
2012 NIMO Problems, 3
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
[i]Proposed by Aaron Lin[/i]
1990 IMO Longlists, 2
The side-lengths of two equilaterals $ABC$ and $KLM$ are $1$ and $1/4$, respectively. And triangle $KLM$ located inside triangle $ABC$. Denote by $\Sigma$ the sum of the distances from $A$ to lines $KL, LM$ and $MK$. Find the location of triangle $KLM$ when $\Sigma$ is maximal.