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For given integer $n \geq 3$ , set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$ . Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$ , one of these number does not lie in between the other two numbers in every permutations $p_i$ ( $1 \leq i \leq m$ ). (For example, in the permutation $(1, 3, 2, 4)$ , $3$ lies in between $1$ and $4$ , and $4$ does not lie in between $1$ and $2$ .) Determine the maximum value of $m$ . | 2^{n-1} | 0.625 |
What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $ | 55 | 0.375 |
Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$ , given that the greatest common factor of $a$ and $b$ is $6$ .
*Proposed by Jacob Xu*
<details><summary>Solution</summary>*Solution*. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$ . Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$ , since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$ .</details> | 186 | 0.875 |
For $n \geq 1$ , let $a_n$ be the number beginning with $n$ $9$ 's followed by $744$ ; eg., $a_4=9999744$ . Define $$ f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \} $$ , for $n\geq 1$ . Find $f(1)+f(2)+f(3)+ \cdots + f(10)$ . | 75 | 0.25 |
Evaluate the following expression: $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$ The terms with minus signs are exactly the powers of two.
| 1996906 | 0.375 |
Let $ABC$ be a triangle in which $\angle ABC = 60^{\circ}$ . Let $I$ and $O$ be the incentre and circumcentre of $ABC$ , respectively. Let $M$ be the midpoint of the arc $BC$ of the circumcircle of $ABC$ , which does not contain the point $A$ . Determine $\angle BAC$ given that $MB = OI$ . | 30^\circ | 0.75 |
Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$ . For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even? | 90 | 0.75 |
Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$ . | 87 | 0.875 |
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$ . In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps? | 1007^2 = 1014049 | 0.25 |
$\textbf{Problem C.1}$ There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ( $A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way? | B | 0.75 |
In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ . | 3 | 0.625 |
Determine the minimum value of $$ x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015 $$ where $x$ is a real number. | 1008 | 0.875 |
Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$ , find the sum of digits of $E$ | 13 | 0.625 |
A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$ , and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel? | 3000 | 0.875 |
Let $r$ be a fixed positive real number. It is known that for some positive integer $n$ the following statement is true: for any positive real numbers $a_1,...,a_n$ satisfying the equation $a_1+...+a_n=r(\frac{1}{a_1}+...+\frac{1}{a_n})$ they also satisfy the equation $\frac{1}{\sqrt{r}-a_1}+...+\frac{1}{\sqrt{r}-a_n}=\frac{1}{\sqrt{r}}$ ( $a_i\neq\sqrt{r}$ ). Find $n$ . | n = 2 | 0.75 |
How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks?
(15 points)
*Proposed by Mostafa Einollah zadeh* | n(k+1) | 0.5 |
A positive integer $N$ greater than $1$ is described as special if in its base- $8$ and base- $9$ representations, both the leading and ending digit of $N$ are equal to $1$ . What is the smallest special integer in decimal representation?
*Proposed by Michael Ren* | 793 | 0.75 |
Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$ , the decimal representation of the number $c^n+2014$ has digits all less than $5$ .
*Proposed by Evan Chen* | 10 | 0.625 |
Triangle $ABC$ has $AB=10$ , $BC=17$ , and $CA=21$ . Point $P$ lies on the circle with diameter $AB$ . What is the greatest possible area of $APC$ ? | \frac{189}{2} | 0.625 |
Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares.
(A perfect square is a number which can be expressed as $k^2$ , where $k$ is an integer.) | 125 | 0.875 |
Find the last two digits of $\tbinom{200}{100}$ . Express the answer as an integer between $0$ and $99$ . (e.g. if the last two digits are $05$ , just write $5$ .) | 20 | 0.375 |
Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$ , $a_2$ , $\dots$ , $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients.
*Proposed by Yang Liu* | n \ge 4 | 0.375 |
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime. | (2, 14) | 0.5 |
In a tennis tournament there are participants from $n$ different countries. Each team consists of a coach and a player whom should settle in a hotel. The rooms considered for the settlement of coaches are different from players' ones. Each player wants to be in a room whose roommates are **<u>all</u>** from countries which have a defense agreement with the player's country. Conversely, each coach wants to be in a room whose roommates are **<u>all</u>** from countries which don't have a defense agreement with the coach's country. Find the minimum number of the rooms such that we can <u>**always**</u> grant everyone's desire.
*proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi* | n + 1 | 0.875 |
For every positive integer $n$ , define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$ . Show that $f(n)$ is a perfect square if and only if $n=1$ . | n = 1 | 0.75 |
Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$ | n = 2 | 0.25 |
Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$ . She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$ . Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this. | 2022 | 0.875 |
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $ , $ a $ , $ b $ , $ c $ , $ d \neq 3 $ , and $ d \le 50 $ . Find the maximum value of the difference between two of these prime numbers. | 2067 | 0.25 |
Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small.
Nair build triangular figures by following these rules: $\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on. $\bullet$ The central column must be made up exclusively of small parts. $\bullet$ Outside the central column, only large pieces can be placed.
Following the pattern, how many pieces will Nair use to build Figure $20$ ? | 460 | 0.875 |
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number. | k = 3 | 0.875 |
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle. | \frac{1}{8} | 0.875 |
For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$
Find the value of $\frac{I_1}{I_2}.$ | 2 | 0.875 |
Let $$ N=2^5+2^{5^2}+2^{5^3}+\cdots{}+2^{5^{2015}}. $$ Wrotten in the usual decimal form, find the last two digits of the number $N$ .
| 80 | 0.625 |
Let $\omega_1$ be a circle of radius $1$ that is internally tangent to a circle $\omega_2$ of radius $2$ at point $A$ . Suppose $\overline{AB}$ is a chord of $\omega_2$ with length $2\sqrt3$ that intersects $\omega_1$ at point $C\ne A$ . If the tangent line of $\omega_1$ at $C$ intersects $\omega_2$ at points $D$ and $E$ , find $CD^4 + CE^4$ . | 63 | 0.625 |
Find the largest positive real $ k$ , such that for any positive reals $ a,b,c,d$ , there is always:
\[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\] | k = 174960 | 0.25 |
How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different?
*Proposed by Evan Chen* | 84 | 0.25 |
For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$ . Find all positive integers $n$ for which $C(2^n+1)=C(n)$ . | n = 3 | 0.625 |
Let $n$ and $k$ be two integers which are greater than $1$ . Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that
i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$ ;
ii) For any integer $m\in\{1,2,\ldots,n\}$ , we have that $c_1+c_2+\ldots+c_m\le m^k$ .
Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$ . | 1 | 0.625 |
Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$ , and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$ ) of distinct numbers in $S$ , meaning
\[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \]
Show that there is a positive integer $N$ (only depending on $n$ ) such that $T$ contains no arithmetic progression of length $N$ .
*Yang Liu* | N | 0.625 |
A semicircle is erected over the segment $AB$ with center $M$ . Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$ . The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$ . Show that $PM = PS$ holds.
*(Karl Czakler)* | PM = PS | 0.875 |
One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$ . The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$ .
What integer was removed ? | 61 | 0.875 |
Let $n$ be a positive integer. Find the greatest possible integer $m$ , in terms of $n$ , with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\]
*Poland (Tomasz Kobos)* | 2^n | 0.75 |
For a positive integer n, let $w(n)$ denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
$2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n. | 5 | 0.5 |
An $\textrm{alien}$ script has $n$ letters $b_1,b_2,\dots,b_n$ . For some $k<n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$ -words. Such a $k$ -word is considered $\textbf{sacred}$ if:
i. no letter appears twice and,
ii. if a letter $b_i$ appears in the word then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$ ).
For example, if $n = 7$ and $k = 3$ then $b_1b_3b_6, b_3b_1b_6, b_2b_4b_6$ are sacred $3$ -words. On the other hand $b_1b_7b_4, b_2b_2b_6$ are not sacred.
What is the total number of sacred $k$ -words?
Use your formula to find the answer for $n = 10$ and $k = 4$ . | 600 | 0.375 |
Princeton’s Math Club recently bought a stock for $\$ 2 $ and sold it for $ \ $9$ thirteen days later. Given that the stock either increases or decreases by $\$ 1 $ every day and never reached $ \ $0$ , in how many possible ways could the stock have changed during those thirteen days?
| 273 | 0.25 |
Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$ . | 1:2 | 0.5 |
$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most? | 7 | 0.375 |
In triangle $ ABC$ , one has marked the incenter, the foot of altitude from vertex $ C$ and the center of the excircle tangent to side $ AB$ . After this, the triangle was erased. Restore it. | \triangle ABC | 0.25 |
The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$ . What is the maximum possible value of $a + b + c$ ? | 43 | 0.875 |
How many ways are there to make two $3$ -digit numbers $m$ and $n$ such that $n=3m$ and each of six digits $1$ , $2$ , $3$ , $6$ , $7$ , $8$ are used exactly once? | 2 | 0.875 |
All angles of $ABC$ are in $(30,90)$ . Circumcenter of $ABC$ is $O$ and circumradius is $R$ . Point $K$ is projection of $O$ to angle bisector of $\angle B$ , point $M$ is midpoint $AC$ . It is known, that $2KM=R$ . Find $\angle B$ | 60^\circ | 0.875 |
No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem | 1681 | 0.125 |
Let $A$ , $B$ , and $C$ be distinct points on a line with $AB=AC=1$ . Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$ . What is the degree measure of the acute angle formed by lines $EC$ and $BF$ ?
*Ray Li* | 75^\circ | 0.875 |
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$ . | 83 | 0.5 |
Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$ , $$ x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}. $$ *Proposed by Morteza Saghafian* | 4 | 0.25 |
Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$ -table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$ . Determine the largest possible value of $M$ . | 2 | 0.125 |
Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$ . Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$ . Lines $BE$ and $DF$ meet at $G$ , and lines $CF$ and $DE$ meet at $H$ . Given that $AB = 8, AC = 10$ , and $\angle BAC = 60^\circ$ , find the area of $BCHG$ .
*Note: this is a modified version of Premier #2* | 2\sqrt{3} | 0.5 |
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other? | 2026085 | 0.5 |
Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$ . | 209 | 0.5 |
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently. | 4 | 0.5 |
Let $ABC$ have side lengths $3$ , $4$ , and $5$ . Let $P$ be a point inside $ABC$ . What is the minimum sum of lengths of the altitudes from $P$ to the side lengths of $ABC$ ? | \frac{12}{5} | 0.625 |
Determine the largest integer $N$ , for which there exists a $6\times N$ table $T$ that has the following properties: $*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering. $*$ For any two columns $i\ne j$ , there exists a row $r$ such that $T(r,i)= T(r,j)$ . $*$ For any two columns $i\ne j$ , there exists a row $s$ such that $T(s,i)\ne T(s,j)$ .
(Proposed by Gerhard Woeginger, Austria) | 120 | 0.25 |
problem 1 :A sequence is defined by $ x_1 = 1, x_2 = 4$ and $ x_{n+2} = 4x_{n+1} -x_n$ for $n \geq 1$ . Find all natural numbers $m$ such that the number $3x_n^2 + m$ is a perfect square for all natural numbers $n$ | m = 1 | 0.875 |
Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$ , $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers. | 6210001000 | 0.125 |
Let $BCDE$ be a trapezoid with $BE\parallel CD$ , $BE = 20$ , $BC = 2\sqrt{34}$ , $CD = 8$ , $DE = 2\sqrt{10}$ . Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$ , and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$ , and let $X$ be the intersection of $AM$ and $BE$ . If $BX$ can be written as $\frac{a}{b}$ , where $a, b$ are relatively prime positive integers, find $a + b$ | 203 | 0.875 |
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player? | 6 | 0.375 |
A four digit number is called *stutterer* if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers. | 7744 | 0.75 |
Let $n$ be a positive integer. For a positive integer $m$ , we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$ , fi nd the largest positive integer $m$ for which such a partition exists. | n^2 + 2n | 0.125 |
The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\]
Show that $ a_{2008} < 5$ . | a_{2008} < 5 | 0.875 |
David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
*Ray Li* | 30 | 0.75 |
How many distinct triangles $ABC$ are tjere, up to simplilarity, such that the magnitudes of the angles $A, B$ and $C$ in degrees are positive integers and satisfy $$ \cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1 $$ for some positive integer $k$ , where $kC$ does not exceet $360^{\circ}$ ? | 6 | 0.625 |
Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$ . Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ . | 24 | 0.125 |
Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$ . Suppose that $a_1$ , $a_3$ , and $a_6$ also form a geometric sequence. Compute $a_1$ . | 12 | 0.875 |
Find all functions $f : R \to R$ satisfying the conditions:
1. $f (x + 1) \ge f (x) + 1$ for all $x \in R$
2. $f (x y) \ge f (x)f (y)$ for all $x, y \in R$ | f(x) = x | 0.875 |
Points $A$ , $B$ , and $C$ lie in that order on line $\ell$ such that $AB=3$ and $BC=2$ . Point $H$ is such that $CH$ is perpendicular to $\ell$ . Determine the length $CH$ such that $\angle AHB$ is as large as possible. | \sqrt{10} | 0.75 |
Determine the smallest natural number $n =>2$ with the property:
For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$ ,
$1 <=i <j <=n$ , is divisible by 2001. | 30 | 0.75 |
Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$ . What is the remainder when $N$ is divided by 2010? | 503 | 0.125 |
A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into. | 2n | 0.5 |
Let $\overline{AB}$ be a line segment with length $10$ . Let $P$ be a point on this segment with $AP = 2$ . Let $\omega_1$ and $\omega_2$ be the circles with diameters $\overline{AP}$ and $\overline{P B}$ , respectively. Let $XY$ be a line externally tangent to $\omega_1$ and $\omega_2$ at distinct points $X$ and $Y$ , respectively. Compute the area of $\vartriangle XP Y$ . | \frac{16}{5} | 0.875 |
On a chess board ( $8*8$ ) there are written the numbers $1$ to $64$ : on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$ ; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$ ;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and in each column, there are $4$ $\"+\"$ signs and $4$ $\"-\"$ signs. Then, the $64$ numbers are added. Find all the possible values of this sum. | 0 | 0.375 |
Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$ . | 45578899 | 0.375 |
In triangle $\triangle ABC$ , the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$ (they are concurrent at $O$ ). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$ . Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$ . | 94 | 0.625 |
Let $S = \{1,2,3,\ldots,n\}$ . Consider a function $f\colon S\to S$ . A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$ . The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$ .**i)** Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$ .**ii)** Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$ . | 2^k | 0.5 |
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number? | 1001 | 0.125 |
A natural number $k$ is said $n$ -squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$ , the smallest natural $k$ that is $n$ -squared. | 2n^2 - n + 1 | 0.125 |
For a positive integer $n$ , let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$ .
Find $I_1+I_2+I_3+I_4$ .
*1992 University of Fukui entrance exam/Medicine* | \frac{40}{9} | 0.75 |
The diagram below is made up of a rectangle AGHB, an equilateral triangle AFG, a rectangle ADEF, and a parallelogram ABCD. Find the degree measure of ∠ABC. For diagram go to http://www.purplecomet.org/welcome/practice, the 2015 middle school contest, and go to #2 | 60^\circ | 0.375 |
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$ | 2 | 0.5 |
Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$ . Let $c$ be the coefficient of $x^{48}$ in $f(x)$ . When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.) | 60 | 0.5 |
Given a convex quadrilateral $ABCD$ in which $\angle BAC = 20^o$ , $\angle CAD = 60^o$ , $\angle ADB = 50^o$ , and $\angle BDC = 10^o$ . Find $\angle ACB$ . | 80^\circ | 0.125 |
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\] | 5 | 0.5 |
Find all positive integers $n$ such that the product of all positive divisors of $n$ is $24^{240}$ . | n = 24^5 | 0.625 |
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and
\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }a_n+h \text{ otherwise }.\end{array}\]
For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$ ? | h | 0.625 |
Find the smallest positive integer $ n$ such that $ 107n$ has the same last two digits as $ n$ . | 50 | 0.875 |
Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$ ,
(i) $(x + 1)\star 0 = (0\star x) + 1$ (ii) $0\star (y + 1) = (y\star 0) + 1$ (iii) $(x + 1)\star (y + 1) = (x\star y) + 1$ .
If $123\star 456 = 789$ , find $246\star 135$ . | 579 | 0.625 |
Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_1$ , $a_2$ , $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$ . | P(x) = ax + b | 0.125 |
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.
*Proposed by DVDthe1st, mzy and jjax* | \frac{2}{3} | 0.5 |
Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$ , one containing M and the other contained in M.
*A. Khrabrov* | \sqrt{2} | 0.625 |
p1. A fraction is called Toba- $n$ if the fraction has a numerator of $1$ and the denominator of $n$ . If $A$ is the sum of all the fractions of Toba- $101$ , Toba- $102$ , Toba- $103$ , to Toba- $200$ , show that $\frac{7}{12} <A <\frac56$ .
p2. If $a, b$ , and $c$ satisfy the system of equations $$ \frac{ab}{a+b}=\frac12 $$ $$ \frac{bc}{b+c}=\frac13 $$ $$ \frac{ac}{a+c}=\frac17 $$ Determine the value of $(a- c)^b$ .
p3. Given triangle $ABC$ . If point $M$ is located at the midpoint of $AC$ , point $N$ is located at the midpoint of $BC$ , and the point $P$ is any point on $AB$ . Determine the area of the quadrilateral $PMCN$ .
p4. Given the rule of motion of a particle on a flat plane $xy$ as following: $N: (m, n)\to (m + 1, n + 1)$ $T: (m, n)\to (m + 1, n - 1)$ , where $m$ and $n$ are integers.
How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ?
p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner. | 6 | 0.125 |
Let $ABC$ be an equilateral triangle of area $1998$ cm $^2$ . Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$ , respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$ , and the line $BM$ intersects the line $CK$ at point $R$ . Find the area of the triangle $PQR$ . | 54 \, \text{cm}^2 | 0.5 |
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