Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
900 |
At least $ n - 1$ numbers are removed from the set $\{1, 2, \ldots, 2n - 1\}$ according to the following rules:
(i) If $ a$ is removed, so is $ 2a$ ;
(ii) If $ a$ and $ b$ are removed, so is $ a \plus{} b$ .
Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.
|
n^2
| 17.96875 |
901 |
The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$ . Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$ , we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$ .
|
\frac{n+1}{n-1}
| 0 |
902 |
A *strip* is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$ . Let $A'$ be a rotation of $A$ in the plane by $60^\circ$ . The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$ . Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$ . What is the greatest possible value of the number $x$ ?
|
150^\circ
| 25.78125 |
903 |
Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs $(i, j)$ , $1\leq i,j\leq 2022$ , satisfying $$ a_i^2+a_j\ge \frac 1{2021}. $$
|
2021 \times 2022
| 1.5625 |
904 |
5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$ . (We include 1 in the set $S$ .) If $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$ , find $a+b$ . (Here $\varphi$ denotes Euler's totient function.)
|
1537
| 0 |
905 |
The expressions $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$ and $ abc$ are called the elementary symmetric expressions on the three letters $ a, b, c;$ symmetric because if we interchange any two letters, say $ a$ and $ c,$ the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let $ S_k(n)$ denote the elementary expression on $ k$ different letters of order $ n;$ for example $ S_4(3) \equal{} abc \plus{} abd \plus{} acd \plus{} bcd.$ There are four terms in $ S_4(3).$ How many terms are there in $ S_{9891}(1989)?$ (Assume that we have $ 9891$ different letters.)
|
\binom{9891}{1989}
| 46.09375 |
906 |
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$ , let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends $X'$ and $X''$ , we have $t(X') \neq t(X''),$ $2)$ For every contestant $X$ , the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?
|
2
| 60.15625 |
907 |
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?
|
6
| 24.21875 |
908 |
Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$ .
|
\frac{1}{4}
| 75.78125 |
909 |
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction).
|
168
| 45.3125 |
910 |
Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)
|
93
| 0.78125 |
911 |
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$ |S| +\ min(S) \cdot \max (S) = 0. $$
|
335
| 58.59375 |
912 |
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$ s that may occur among the $100$ numbers.
|
95
| 1.5625 |
913 |
Let $ y_0$ be chosen randomly from $ \{0, 50\}$ , let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$ , let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$ , and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$ , $ P(1) \equal{} y_1$ , $ P(2) \equal{} y_2$ , and $ P(3) \equal{} y_3$ . What is the expected value of $ P(4)$ ?
|
107
| 1.5625 |
914 |
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$ .
What is the area of triangle $ABC$ ?
|
200
| 43.75 |
915 |
To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$ . Solve the equation $x*36 = 216$ .
|
7776
| 75 |
916 |
How many ways are there to insert $+$ 's between the digits of $111111111111111$ (fifteen $1$ 's) so that the result will be a multiple of $30$ ?
|
2002
| 0.78125 |
917 |
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For example, if she cuts the $1111$ th link out of her chain first, then she will have $3$ chains, of lengths $1110$ , $1$ , and $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from $1$ to $2018$ links using some combination of her chains?
*2018 CCA Math Bonanza Individual Round #10*
|
10
| 61.71875 |
918 |
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$
|
9
| 60.9375 |
919 |
In an acute scalene triangle $ABC$ , points $D,E,F$ lie on sides $BC, CA, AB$ , respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$ . Altitudes $AD, BE, CF$ meet at orthocenter $H$ . Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$ . Lines $DP$ and $QH$ intersect at point $R$ . Compute $HQ/HR$ .
*Proposed by Zuming Feng*
|
1
| 65.625 |
920 |
For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$
|
19
| 57.8125 |
921 |
Find a costant $C$ , such that $$ \frac{S}{ab+bc+ca}\le C $$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.)
|
\frac{1}{4\sqrt{3}}
| 4.6875 |
922 |
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
|
5
| 72.65625 |
923 |
Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$
|
1
| 99.21875 |
924 |
Evaluate $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$ , where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$ .
|
\frac{\pi}{2}
| 90.625 |
925 |
Construct the $ \triangle ABC$ , given $ h_a$ , $ h_b$ (the altitudes from $ A$ and $ B$ ) and $ m_a$ , the median from the vertex $ A$ .
|
\triangle ABC
| 3.90625 |
926 |
Let $a_1=24$ and form the sequence $a_n$ , $n\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ?
|
88
| 26.5625 |
927 |
Find the number of ordered quadruples of positive integers $(a,b,c, d)$ such that $ab + cd = 10$ .
|
58
| 92.96875 |
928 |
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$ . Find the maximum possible value of $A \cdot B$ .
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$ .
|
143
| 1.5625 |
929 |
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique **alternating sum** is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$ .
|
448
| 87.5 |
930 |
A four-digit positive integer is called *virtual* if it has the form $\overline{abab}$ , where $a$ and $b$ are digits and $a \neq 0$ . For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$ , for some positive integer $n$ .
|
8282
| 100 |
931 |
Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$ . Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$ , what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of $k$ ?
*2018 CCA Math Bonanza Lightning Round #4.1*
|
1009k
| 9.375 |
932 |
On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$ . He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$ . Find the difference between largest and smallest values Donald can achieve.
|
0
| 20.3125 |
933 |
Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by
\[\tau(f)(x)=f(x+1).\]
Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$ . Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$ .
|
g = x^{p-1}
| 0 |
934 |
The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$ . Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$ ? Point A on the board located above and to the left of point $B$ .
|
O(0, 0)
| 11.71875 |
935 |
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
|
n = 3
| 0 |
936 |
Pick out three numbers from $0,1,\cdots,9$ , their sum is an even number and not less than $10$ . We have________different ways to pick numbers.
|
51
| 98.4375 |
937 |
If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$ , what is the probability that $|a|+|b|<1$ ?
|
\frac{1}{2}
| 64.84375 |
938 |
a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$ .
b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.
|
\det(A) > 0
| 33.59375 |
939 |
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$ , compute $P(1)$ .
|
100
| 1.5625 |
940 |
Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes.
|
175
| 100 |
941 |
Let $n\in \mathbb{Z}_{> 0}$ . The set $S$ contains all positive integers written in decimal form that simultaneously satisfy the following conditions:
[list=1][*] each element of $S$ has exactly $n$ digits;
[*] each element of $S$ is divisible by $3$ ;
[*] each element of $S$ has all its digits from the set $\{3,5,7,9\}$ [/list]
Find $\mid S\mid$
|
\frac{4^n + 2}{3}
| 8.59375 |
942 |
Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$ . Find all possible values of $x+y$ .
|
0, 3, 4, 5, 8
| 35.15625 |
943 |
Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$ , then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty.
|
K = 1011
| 0 |
944 |
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$ , such that $a_j = a_i + 1$ and $a_k = a_j + 1$ . Considering all such sequences $A$ , find the greatest value of $m$ .
|
667^3
| 0 |
945 |
How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ?
|
120
| 51.5625 |
946 |
Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.
|
n = 3
| 0 |
947 |
In triangle $ABC$ , let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$ . Find $OH$ .
*Proposed by Kevin You*
|
10
| 6.25 |
948 |
A positive integer is called *oneic* if it consists of only $1$ 's. For example, the smallest three oneic numbers are $1$ , $11$ , and $111$ . Find the number of $1$ 's in the smallest oneic number that is divisible by $63$ .
|
18
| 45.3125 |
949 |
Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$ , then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$ .
|
n
| 10.15625 |
950 |
Find the least positive integer $n$ such that the prime factorizations of $n$ , $n + 1$ , and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).
|
33
| 99.21875 |
951 |
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists.
|
5
| 58.59375 |
952 |
Let $f(x)$ be a function such that $f(1) = 1234$ , $f(2)=1800$ , and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$ . Evaluate the number of divisors of
\[\sum_{i=1}^{2022}f(i)\]
*2022 CCA Math Bonanza Tiebreaker Round #4*
|
8092
| 1.5625 |
953 |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer.
|
725
| 85.15625 |
954 |
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.
|
130
| 94.53125 |
955 |
Find all triplets $ (x,y,z) $ of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \]
|
(6, 2, 10)
| 7.03125 |
956 |
Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?
|
\frac{8}{17}
| 0.78125 |
957 |
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$ , it is true that at least one of the following numbers: $$ a, b,\frac{5}{a^2}+\frac{6}{b^3} $$ is less than or equal to $k$ .
|
2
| 32.8125 |
958 |
Positive sequences $\{a_n\},\{b_n\}$ satisfy: $a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$ .
Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$ ,where $m$ is a given positive integer.
|
5
| 28.90625 |
959 |
Leo the fox has a $5$ by $5$ checkerboard grid with alternating red and black squares. He fills in the
grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares
(sharing a side) and each number is used exactly once. He then computes the sum of the numbers in
the $13$ squares that are the same color as the center square. Compute the maximum possible sum Leo
can obtain.
|
169
| 50.78125 |
960 |
Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let $f^{(k)}$ denote the $k$ th derivative of $f$ , compute $f^{(10)}(1)$ . For the sake of this problem, note that $10!=3628800$ .
|
907200
| 0 |
961 |
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.
|
49
| 28.90625 |
962 |
Jane and Josh wish to buy a candy. However Jane needs seven more cents to buy the candy, while John needs one more cent. They decide to buy only one candy together, but discover that they do not have enough money. How much does the candy cost?
|
7
| 49.21875 |
963 |
If the rotational inertia of a sphere about an axis through the center of the sphere is $I$ , what is the rotational inertia of another sphere that has the same density, but has twice the radius? $ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $
|
32I
| 89.84375 |
964 |
Let $ABCD$ be a square of side length $1$ , and let $E$ and $F$ be points on $BC$ and $DC$ such that $\angle{EAF}=30^\circ$ and $CE=CF$ . Determine the length of $BD$ .
*2015 CCA Math Bonanza Lightning Round #4.2*
|
\sqrt{2}
| 97.65625 |
965 |
Alex starts with a rooted tree with one vertex (the root). For a vertex $v$ , let the size of the subtree of $v$ be $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex $v_1$ ). The expected value of $S(v_1)$ is of the form $\tfrac{m}{n}$ for relatively prime positive integers $m, n$ . Find $m+n$ .**Note:** In a rooted tree, the subtree of $v$ consists of its indirect or direct descendants (including $v$ itself).
*Proposed by Yang Liu*
|
9901
| 0.78125 |
966 |
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign " $+$ " or a multiplication sign " $\times$ " between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$ , with value 13.) Each positive digit is equally likely, each arithmetic sign (" $+$ " or " $\times$ ") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
|
50
| 36.71875 |
967 |
Let $B_n$ be the set of all sequences of length $n$ , consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$ \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1). $$ . Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $ .
.
|
n \cdot 4^{n-1}
| 2.34375 |
968 |
(a) Find the smallest number of lines drawn on the plane so that they produce exactly 2022 points of intersection. (Note: For 1 point of intersection, the minimum is 2; for 2 points, minimum is 3; for 3 points, minimum is 3; for 4 points, minimum is 4; for 5 points, the minimum is 4, etc.)
(b) What happens if the lines produce exactly 2023 intersections?
|
k = 65
| 0 |
969 |
In the quadrilateral $ABCD$ , we have $\measuredangle BAD = 100^{\circ}$ , $\measuredangle BCD = 130^{\circ}$ , and $AB=AD=1$ centimeter. Find the length of diagonal $AC$ .
|
1 \text{ cm}
| 16.40625 |
970 |
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?
Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$ .
|
13
| 31.25 |
971 |
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$ . How many progressions does Omar's list have?
|
6
| 64.0625 |
972 |
Let $ABC$ be a triangle with $|AB|=|AC|=26$ , $|BC|=20$ . The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$ , respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$ .
|
\frac{65}{12}
| 0 |
973 |
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by *Nikola Velov, Macedonia*
|
100
| 58.59375 |
974 |
Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine $ \angle{AQC}$ .
|
45^\circ
| 92.1875 |
975 |
Let $p$ be a permutation of the set $S_n = \{1, 2, \dots, n\}$ . An element $j \in S_n$ is called a fixed point of $p$ if $p(j) = j$ . Let $f_n$ be the number of permutations having no fixed points, and $g_n$ be the number with exactly one fixed point. Show that $|f_n - g_n| = 1$ .
|
|f_n - g_n| = 1
| 0 |
976 |
In triangle $ABC$ , side $AB$ has length $10$ , and the $A$ - and $B$ -medians have length $9$ and $12$ , respectively. Compute the area of the triangle.
*Proposed by Yannick Yao*
|
72
| 3.90625 |
977 |
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals $5$ red chips, $3$ red tile are equal to $ 8$ black tiles, and a black tile is worth $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to $560$ and there are no more than five tiles of the same color.
|
3
| 96.875 |
978 |
Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$ .
How many solutions (including University Mathematics )are there for the problem?
Any advice would be appreciated. :)
|
\frac{1}{8}
| 55.46875 |
979 |
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$ , and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$ ?
|
578
| 0 |
980 |
Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ( $1 \times 1$ squares) are *neighbors* if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors.
An integer from 1 to 4 is written inside each square according to the following rules:
- If a cell has a 2 written on it, then at least two of its neighbors contain a 1.
- If a cell has a 3 written on it, then at least three of its neighbors contain a 1.
- If a cell has a 4 written on it, then all of its neighbors contain a 1.
Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?
|
\frac{5n^2}{2}
| 55.46875 |
981 |
In $\triangle ABC$ , $AB = 40$ , $BC = 60$ , and $CA = 50$ . The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$ . Find $BP$ .
*Proposed by Eugene Chen*
|
40
| 36.71875 |
982 |
Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$ , both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$ .
|
865
| 15.625 |
983 |
Find all real numbers $x,y,z$ such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$
|
x = y = z = 1
| 0 |
984 |
In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?
|
21
| 20.3125 |
985 |
For $k=1,2,\dots$ , let $f_k$ be the number of times
\[\sin\left(\frac{k\pi x}{2}\right)\]
attains its maximum value on the interval $x\in[0,1]$ . Compute
\[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]
|
\frac{1}{4}
| 100 |
986 |
You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that none of the digits are prime, 0, or 1, and that the average value of the digits is 5.
How many combinations will you have to try?
|
10
| 91.40625 |
987 |
A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ( $k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$ <span style="font-size:75%">1</span> and $R$ <span style="font-size:75%">2</span> there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$ <span style="font-size:75%">1</span> with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$ <span style="font-size:75%">2</span> with the columns from $S$ there are odd number of $1's$ .
Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.
|
n
| 92.1875 |
988 |
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$ , then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$ .
|
96
| 3.125 |
989 |
Let $S$ be the set of all real numbers $x$ such that $0 \le x \le 2016 \pi$ and $\sin x < 3 \sin(x/3)$ . The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals.
|
1008\pi
| 33.59375 |
990 |
At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions left (for example, the list 2, 3, 4, 6 closes right after we add 12). Which is the maximum numbers which can be written on a list that had closed, if the list had at the beginning 10 numbers?
|
2^{10} - 1 = 1023
| 0 |
991 |
Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$ , what is the area of the trapezoid?
|
100\sqrt{3}
| 15.625 |
992 |
For a positive integer $n$ , define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer $k$ for which $7?9?=5?k?$ .
*Proposed by Tristan Shin*
|
10
| 22.65625 |
993 |
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
*2019 CCA Math Bonanza Individual Round #5*
|
40
| 20.3125 |
994 |
What is the least posssible number of cells that can be marked on an $n \times n$ board such that for each $m >\frac{ n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?
|
n
| 12.5 |
995 |
The **Collaptz function** is defined as $$ C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \frac{n}{2} & n\textrm{~even}.\end{cases} $$
We obtain the **Collaptz sequence** of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. Find the sum of the three smallest positive integers $n$ whose Collaptz sequences do not contain $1,$ or in other words, do not **collaptzse**.
*Proposed by Andrew Wu and Jason Wang*
|
21
| 54.6875 |
996 |
Find all positive integers $(m, n)$ that satisfy $$ m^2 =\sqrt{n} +\sqrt{2n + 1}. $$
|
(13, 4900)
| 35.15625 |
997 |
Find the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} . $$
|
f(x) = x
| 84.375 |
998 |
Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \]*Proposed by Aaron Lin*
|
504
| 87.5 |
999 |
Find all positive integers $k$ satisfying: there is only a finite number of positive integers $n$ , such that the positive integer solution $x$ of $xn+1\mid n^2+kn+1$ is not unique.
|
k \neq 2
| 0 |
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