Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
1,200 |
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a *box*. Two boxes *intersect* if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$ , $ \ldots$ , $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$ .
*Proposed by Gerhard Woeginger, Netherlands*
|
6
| 7.03125 |
1,201 |
If $(2^x - 4^x) + (2^{-x} - 4^{-x}) = 3$ , find the numerical value of the expression $$ (8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}). $$
|
-1
| 3.90625 |
1,202 |
Given a positive integer $c$ , we construct a sequence of fractions $a_1, a_2, a_3,...$ as follows: $\bullet$ $a_1 =\frac{c}{c+1} $ $\bullet$ to get $a_n$ , we take $a_{n-1}$ (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add $2$ to the numerator and $3$ to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator.
For example, if we take $c = 20$ , then $a_1 =\frac{20}{21}$ and $a_2 =\frac{22}{24} = \frac{11}{12}$ . Then we find that $a_3 =\frac{13}{15}$ (which is already simplified) and $a_4 =\frac{15}{18} =\frac{5}{6}$ .
(a) Let $c = 10$ , hence $a_1 =\frac{10}{11}$ . Determine the largest $n$ for which a simplification is needed in the construction of $a_n$ .
(b) Let $c = 99$ , hence $a_1 =\frac{99}{100}$ . Determine whether a simplification is needed somewhere in the sequence.
(c) Find two values of $c$ for which in the first step of the construction of $a_5$ (before simplification) the numerator and denominator are divisible by $5$ .
|
n = 7
| 0 |
1,203 |
Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$ , $\overline{BE}$ , and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$ , the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$ .
|
5
| 53.125 |
1,204 |
The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$ th page of this question, you write the fraction $i/p$ to denote that this is the $i$ th page out of $p$ for this question. When you turned in your submissions for the $2017$ USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be $2017$ . How many pages did you turn in?
*Proposed by Tristan Shin*
|
4028
| 53.90625 |
1,205 |
Let $P$ be a regular $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$ . The sides of $P$ are also called *good*.
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$ . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
|
1003
| 90.625 |
1,206 |
The sum $$ \frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!} $$ $ $
can be expressed as a rational number $N$ . Find the last 3 digits of $2021! \cdot N$ .
|
977
| 12.5 |
1,207 |
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$ , where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$ .
|
821
| 0 |
1,208 |
Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ( $k\ge1$ ) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$ , exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$ .
*David Yang.*
|
(m, n)
| 0.78125 |
1,209 |
The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$ , calculate the area of triangle $PQR$ .
|
\frac{9}{2}
| 0 |
1,210 |
For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?
|
3
| 31.25 |
1,211 |
Show that among the vertices of any area $1$ convex polygon with $n > 3$ sides there exist four such that the quadrilateral formed by these four has area at least $1/2$ .
|
\frac{1}{2}
| 73.4375 |
1,212 |
Suppose that for some positive integer $n$ , the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$ .
|
31
| 84.375 |
1,213 |
Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$ , $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$ , find $a + b$ .
*Proposed by Andrew Wu*
|
24
| 51.5625 |
1,214 |
Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$ , there exists an integer $r$ with $0 \le r \le n - k$ which satisfies
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]
|
2^{n-1}
| 10.9375 |
1,215 |
Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and at least $2$ of those $20$ don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any $social$ nor $shy$ persons.
|
40
| 17.1875 |
1,216 |
The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
|
52
| 14.0625 |
1,217 |
An *up-right path* from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$ , and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$ . Two up-right paths are said to intersect if they share any point.
Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$ , $B$ is an up-right path from $(2, 0)$ to $(6, 4)$ , and $A$ and $B$ do not intersect.
|
1750
| 2.34375 |
1,218 |
Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$ .
|
\alpha = 8
| 0 |
1,219 |
Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$ , the value of $$ \sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} $$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ .
|
329
| 1.5625 |
1,220 |
Find all pairs $(a,b)$ of positive integers, such that for **every** $n$ positive integer, the equality $a^n+b^n=c_n^{n+1}$ is true, for some $c_n$ positive integer.
|
(2, 2)
| 46.09375 |
1,221 |
Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$ .
*Proposed by David Tang*
|
10
| 95.3125 |
1,222 |
$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$ , and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$ , such that
\[
p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1
\]
Show that $p'(x) = nq(x)$ .
|
p'(x) = nq(x)
| 87.5 |
1,223 |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$ 's. If a number is chosen at random from $S$ , the probability that it is divisible by $9$ is $p/q$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .
|
913
| 35.15625 |
1,224 |
Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . The incircle of $ABC$ meets $BC$ at $D$ . Line $AD$ meets the circle through $B$ , $D$ , and the reflection of $C$ over $AD$ at a point $P\neq D$ . Compute $AP$ .
*2020 CCA Math Bonanza Tiebreaker Round #4*
|
2\sqrt{145}
| 0 |
1,225 |
Define a sequence $(a_n)$ recursively by $a_1=0, a_2=2, a_3=3$ and $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$ for $n \ge 4$ . Determine the prime factorization of $a_{19702020}$ .
|
3^{6567340}
| 0 |
1,226 |
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5,$ $b_1 = 7,$ $$ a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2}, $$ and $$ b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}. $$ $ $
How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?
|
24
| 1.5625 |
1,227 |
The *rank* of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$ , where each $a_i$ is the smallest positive integer $q$ such that $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$ . Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank $3$ , and suppose the expression for $q$ is $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$ . Find the ordered triple $(a_1,a_2,a_3)$ .
|
(5, 21, 421)
| 77.34375 |
1,228 |
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$ . Point $D$ is not on $\overline{AC}$ so that $AD = CD$ , and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$ . Find $s$ .
|
380
| 0.78125 |
1,229 |
Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$ , $y$ , $z$ are nonnegative real numbers such that $x+y+z=1$ .
|
\frac{7}{18}
| 60.9375 |
1,230 |
For $n\geq 2$ , an equilateral triangle is divided into $n^2$ congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the $\frac{(n+1)(n+2)}{2}$ vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.
|
0
| 67.1875 |
1,231 |
Bobbo starts swimming at $2$ feet/s across a $100$ foot wide river with a current of $5$ feet/s. Bobbo doesn’t know that there is a waterfall $175$ feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side of the river safely?
|
3
| 32.8125 |
1,232 |
Find all functions $f:Q\rightarrow Q$ such that \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \] for all $x,y,z,t\in Q.$
|
f(x) = c
| 46.875 |
1,233 |
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$ , write the square and the cube of a positive integer. Determine what that number can be.
|
24
| 74.21875 |
1,234 |
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 $ .
|
150^\circ
| 93.75 |
1,235 |
Let $f(x)=\sum_{i=1}^{2014}|x-i|$ . Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval.
|
1
| 99.21875 |
1,236 |
Find maximum of the expression $(a -b^2)(b - a^2)$ , where $0 \le a,b \le 1$ .
|
\frac{1}{16}
| 58.59375 |
1,237 |
Let $a$ and $b$ be real numbers such that $ \frac {ab}{a^2 + b^2} = \frac {1}{4} $ . Find all possible values of $ \frac {|a^2-b^2|}{a^2+b^2} $ .
|
\frac{\sqrt{3}}{2}
| 88.28125 |
1,238 |
Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points.
(Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$ )
|
21
| 95.3125 |
1,239 |
Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions:
(a) $P(2017)=2016$ and
(b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$ .
|
P(x) = x - 1
| 91.40625 |
1,240 |
Find the least real number $k$ with the following property: if the real numbers $x$ , $y$ , and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]
|
\frac{16}{9}
| 34.375 |
1,241 |
On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?
|
n
| 93.75 |
1,242 |
Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.
|
n = 2
| 0 |
1,243 |
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$ $(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$ *In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.*
|
k = 3
| 0 |
1,244 |
A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$ , then $ab\not\in X$ . What is the maximal number of elements in $X$ ?
|
9901
| 56.25 |
1,245 |
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
|
0, 50, 100, 101
| 0 |
1,246 |
Let $ AB$ be the diameter of a circle with a center $ O$ and radius $ 1$ . Let $ C$ and $ D$ be two points on the circle such that $ AC$ and $ BD$ intersect at a point $ Q$ situated inside of the circle, and $ \angle AQB\equal{} 2 \angle COD$ . Let $ P$ be a point that intersects the tangents to the circle that pass through the points $ C$ and $ D$ .
Determine the length of segment $ OP$ .
|
\frac{2\sqrt{3}}{3}
| 0.78125 |
1,247 |
Find the number of subsets $ S$ of $ \{1,2, \dots 63\}$ the sum of whose elements is $ 2008$ .
|
6
| 100 |
1,248 |
We call a number greater than $25$ , *semi-prime* if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be *semi-prime*?
|
5
| 97.65625 |
1,249 |
The triangle $ABC$ is $| BC | = a$ and $| AC | = b$ . On the ray starting from vertex $C$ and passing the midpoint of side $AB$ , choose any point $D$ other than vertex $C$ . Let $K$ and $L$ be the projections of $D$ on the lines $AC$ and $BC$ , respectively, $K$ and $L$ . Find the ratio $| DK | : | DL |$ .
|
\frac{a}{b}
| 17.1875 |
1,250 |
A kite is inscribed in a circle with center $O$ and radius $60$ . The diagonals of the kite meet at a point $P$ , and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$ , where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$ .
|
239
| 1.5625 |
1,251 |
Let $a_0$ , $a_1$ , $a_2$ , $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$ . Let $c$ be the smallest number such that for every positive integer $n$ , the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$ , rounded to the nearest integer?
|
167
| 8.59375 |
1,252 |
Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$ . The resulting three smaller triangles have areas $1$ , $4$ , and $9$ . Find the area of $\triangle ABC$ .
[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }
pair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);
draw(A--B--C--cycle);
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);
filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));
filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));
filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]
|
36
| 55.46875 |
1,253 |
A *T-tetromino* is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.
|
4
| 14.84375 |
1,254 |
The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.
|
249001
| 57.03125 |
1,255 |
In a tetrahedral $ABCD$ , given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$ , $AD=BD=3$ , and $CD=2$ . Find the radius of the circumsphere of $ABCD$ .
|
\sqrt{3}
| 8.59375 |
1,256 |
Find all $10$ -digit whole numbers $N$ , such that first $10$ digits of $N^2$ coincide with the digits of $N$ (in the same order).
|
1000000000
| 10.15625 |
1,257 |
Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$
|
20
| 57.8125 |
1,258 |
Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying
\[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0,
1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases}
\]
Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.
|
f(p, q) = pq
| 0.78125 |
1,259 |
There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all of them were in the shop at a speci c time instance or no two of them were both in the shop at any time instance.
|
45
| 4.6875 |
1,260 |
Let $T$ be the answer to question $18$ . Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$ . Point $B$ lies on segment $ZO$ , $O'$ lies on segment $OM$ , and $E$ lies on segment $RM$ such that $BR = BE = EO'$ , and $\angle BEO' = 90^o$ . Compute $2(ZO + O'M + ER)$ .
PS. You had better calculate it in terms of $T$ .
|
7T
| 2.34375 |
1,261 |
Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24$ , $q(0)=30$ , and \[p(q(x))=q(p(x))\] for all real numbers $x$ . Find the ordered pair $(p(3),q(6))$ .
|
(3, -24)
| 0.78125 |
1,262 |
One of the receipts for a math tournament showed that $72$ identical trophies were purchased for $\$ $-$ 99.9$-, where the first and last digits were illegible. How much did each trophy cost?
|
11.11
| 7.8125 |
1,263 |
A wall made of mirrors has the shape of $\triangle ABC$ , where $AB = 13$ , $BC = 16$ , and $CA = 9$ . A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$ . The shot reflects about $BC$ and then strikes point $P$ on $AB$ . If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$ , compute $100m+n$ .
*Proposed by Michael Tang*
|
2716
| 0.78125 |
1,264 |
Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$ . Find the sum of all the possible values of $n$ in these ordered pairs.
|
237
| 28.125 |
1,265 |
Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$ . A square $ABCD$ is constructed such that $A$ lies on $L_1$ , $B$ lies on $L_3$ and $C$ lies on $L_2$ . Find the area of the square.
|
45
| 1.5625 |
1,266 |
Suppose that the roots of the quadratic $x^2 + ax + b$ are $\alpha$ and $\beta$ . Then $\alpha^3$ and $\beta^3$ are the roots of some quadratic $x^2 + cx + d$ . Find $c$ in terms of $a$ and $b$ .
|
a^3 - 3ab
| 98.4375 |
1,267 |
The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$ . It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$ . Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$ .
|
24
| 89.0625 |
1,268 |
A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$ . The wheel is perfectly circular and has radius $5$ . After the three laps, how many revolutions around its axis has the wheel been turned through?
|
21
| 99.21875 |
1,269 |
Two 10-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.
|
9 \cdot 10^8
| 0 |
1,270 |
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$ , whose apex $F$ is on the leg $AC$ . Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$ ?
|
2:3
| 67.96875 |
1,271 |
Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice,
where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many people are in Cambridge?
|
7
| 79.6875 |
1,272 |
A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.
How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic?
(Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as
\[A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}\]
respectively, where $H(a,b)$ is not defined for some $a$ , $b$ .)
|
1004
| 0.78125 |
1,273 |
In $\triangle ABC$ , $AB= 425$ , $BC=450$ , and $AC=510$ . An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$ , find $d$ .
|
306
| 65.625 |
1,274 |
Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$ .
|
(1, 1)
| 99.21875 |
1,275 |
Let $p$ be a prime number of the form $4k+1$ . Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]
|
\frac{p-1}{2}
| 99.21875 |
1,276 |
Positive real numbers $a$ , $b$ , $c$ satisfy $a+b+c=1$ . Find the smallest possible value of $$ E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}. $$
|
\frac{1}{8}
| 100 |
1,277 |
Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $ **a)** Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $ **b)** Find the biggest real number $ a $ for which the following inequality is true: $$ \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} . $$
|
2
| 95.3125 |
1,278 |
Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called *joke* if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the number of joke sequences in terms of $n$ .
|
4n^3
| 0 |
1,279 |
On the coordinate plane is given the square with vertices $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$ . For every $n\in\mathbb N$ , point $T_{n+4}$ is defined as the midpoint of the segment $T_nT_{n+1}$ . Determine the coordinates of the limit point of $T_n$ as $n\to\infty$ , if it exists.
|
(0, 0)
| 95.3125 |
1,280 |
A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes.
The grey stripes in the quadrilateral have a combined area of $10$ . Determine the area of the quadrilateral.
|
20
| 100 |
1,281 |
The world-renowned Marxist theorist *Joric* is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$ , he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the *defect* of the number $n$ . Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$ , compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\]
*Iurie Boreico*
|
\frac{1}{2}
| 32.03125 |
1,282 |
For every positive integeer $n>1$ , let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$ .
Find $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$
|
1
| 75.78125 |
1,283 |
Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$ .
|
6
| 12.5 |
1,284 |
The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$ , where $n$ and $k$ are integers and $0\leq k<2013$ . What is $k$ ? Recall $2013=3\cdot 11\cdot 61$ .
|
27
| 45.3125 |
1,285 |
The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?
|
14
| 60.15625 |
1,286 |
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.
|
25
| 10.15625 |
1,287 |
If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$ , then determine $|a + b|$ .
|
3 \sqrt{3}
| 45.3125 |
1,288 |
Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$ .
|
505
| 22.65625 |
1,289 |
**Problem 2**
Determine all pairs $(n, m)$ of positive integers satisfying the equation $$ 5^n = 6m^2 + 1\ . $$
|
(2, 2)
| 4.6875 |
1,290 |
For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ .
|
1680
| 57.03125 |
1,291 |
Triangle $ABC$ has a right angle at $C$ , and $D$ is the foot of the altitude from $C$ to $AB$ . Points $L, M,$ and $N$ are the midpoints of segments $AD, DC,$ and $CA,$ respectively. If $CL = 7$ and $BM = 12,$ compute $BN^2$ .
|
193
| 14.0625 |
1,292 |
For every n = 2; 3; : : : , we put $$ A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right) $$ Determine all positive integer $ n (n \geq 2)$ such that $\frac{1}{A_n}$ is an integer.
|
n=4
| 44.53125 |
1,293 |
How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$ ?
|
3^n
| 90.625 |
1,294 |
A week ago, Sandy’s seasonal Little League batting average was $360$ . After five more at bats this week, Sandy’s batting average is up to $400$ . What is the smallest number of hits that Sandy could have had this season?
|
12
| 80.46875 |
1,295 |
Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$ . Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]
|
c = n
| 0 |
1,296 |
At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar.
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?
(A statement that is at least partially false is considered false.)
|
9
| 1.5625 |
1,297 |
Integer $n>2$ is given. Find the biggest integer $d$ , for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$ .
|
n-1
| 7.8125 |
1,298 |
Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$ .
|
y = x^2 + 1
| 60.9375 |
1,299 |
Let $ p \geq 3$ be a prime, and let $ p$ points $ A_{0}, \ldots, A_{p-1}$ lie on a circle in that order. Above the point $ A_{1+\cdots+k-1}$ we write the number $ k$ for $ k=1, \ldots, p$ (so $ 1$ is written above $ A_{0}$ ). How many points have at least one number written above them?
|
\frac{p+1}{2}
| 37.5 |
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